Solved: In Exercises 4572, determine the convergence or divergence of the sequence with

In Exercises 1–10, write the first five terms of the sequence. \(a_{n}=3^{n}\) Text Transcription: a_n=3^n

In Exercises 1–10, write the first five terms of the sequence. \(a_{n}=\frac{3^{n}}{n !}\) Text Transcription: a_n=frac3^nn!

In Exercises 1–10, write the first five terms of the sequence. \(a_{n}=\left(-\frac{1}{4}\right)^{n}\) Text Transcription: a_n=(-frac14)^n

In Exercises 1–10, write the first five terms of the sequence. \(a_{n}=\left(-\frac{2}{3}\right)^{n}\) Text Transcription: a_n=(-frac23)^

In Exercises 1–10, write the first five terms of the sequence. \(a_{n}=\sin \frac{n \pi}{2}\) Text Transcription: a_n=sinfracnpi2

In Exercises 1–10, write the first five terms of the sequence. \(a_{n}=\frac{2 n}{n+3}\) Text Transcription: a_n=frac2nn+3

In Exercises 1–10, write the first five terms of the sequence. \(a_{n}=\frac{(-1)^{n(n+1) / 2}}{n^{2}}\) Text Transcription: a_n=frac(-1)^n(n+1)/2n^2

In Exercises 1–10, write the first five terms of the sequence. \(a_{n}=(-1)^{n+1}\left(\frac{2}{n}\right)\) Text Transcription: a_n=(-1)^n+1t(frac2n)

In Exercises 1–10, write the first five terms of the sequence. \(a_{n}=5-\frac{1}{n}+\frac{1}{n^{2}}\) Text Transcription: a_n=5-frac1n+frac1n^2

In Exercises 1–10, write the first five terms of the sequence. \(a_{n}=10+\frac{2}{n}+\frac{6}{n^{2}}\) Text Transcription: a_n=10+frac2n+frac6n^2

In Exercises 11–14, write the first five terms of the recursively defined sequence. \(a_{1}=3, a_{k+1}=2\left(a_{k}-1\right)\) Text Transcription: a_1=3,a_k+1=2(a_k-1)

In Exercises 11–14, write the first five terms of the recursively defined sequence. \(a_{1}=4, a_{k+1}=\left(\frac{k+1}{2}\right) a_{k}\) Text Transcription: a_1=4,a_k+1=(frack+12)a_k

In Exercises 11–14, write the first five terms of the recursively defined sequence. \(a_{1}=32, a_{k+1}=\frac{1}{2} a_{k}\) Text Transcription: a_1=32, a_k+1=frac12a_k

In Exercises 11–14, write the first five terms of the recursively defined sequence. \(a_{1}=6, a_{k+1}=\frac{1}{3} a_{k}^{2}\) Text Transcription: a_1=6,a_k+1=frac13a_k^2

In Exercises 15–18, match the sequence with its graph. [The graphs are labeled (a), (b), (c), and (d).] \(a_{n}=\frac{10}{n+1}\) Text Transcription: a_n=frac10n+1

In Exercises 15–18, match the sequence with its graph. [The graphs are labeled (a), (b), (c), and (d).] \(a_{n}=\frac{10 n}{n+1}\) Text Transcription: a_n=frac10nn+1

In Exercises 15–18, match the sequence with its graph. [The graphs are labeled (a), (b), (c), and (d).] \(a_{n}=(-1)^{n}\) Text Transcription: a_n=(-1)^n

In Exercises 15–18, match the sequence with its graph. [The graphs are labeled (a), (b), (c), and (d).] \(a_{n}=\frac{(-1)^{n}}{n}\) Text Transcription: a_n=frac(-1)^nn

In Exercises 19–22, match the sequence with the correct expression for its nth term. [The nth terms are labeled (a), (b), (c), and (d).] (a) \(a_{n}=\frac{2}{3} n\) (b) \(a_{n}=2-\frac{4}{n}\) (c) \(a_{n}=16(-0.5)^{n-1}\) (d) \(a_{n}=\frac{2 n}{n+1}\) Text Transcription: a_n=frac23n a_n=2-frac4n a_n=16(-0.5)^n-1 a_n=frac2nn+1

In Exercises 19–22, match the sequence with the correct expression for its nth term. [The nth terms are labeled (a), (b), (c), and (d).] (a) \(a_{n}=\frac{2}{3} n\) (b) \(a_{n}=2-\frac{4}{n}\) (c) \(a_{n}=16(-0.5)^{n-1}\) (d) \(a_{n}=\frac{2 n}{n+1}\) \(16,-8,4,-2, \ldots\) Text Transcription: 16,-8,4,-2,ldots

In Exercises 19–22, match the sequence with the correct expression for its nth term. [The nth terms are labeled (a), (b), (c), and (d).] (a) \(a_{n}=\frac{2}{3} n\) (b) \(a_{n}=2-\frac{4}{n}\) (c) \(a_{n}=16(-0.5)^{n-1}\) (d) \(a_{n}=\frac{2 n}{n+1}\) \(\frac{2}{3}, \frac{4}{3}, 2, \frac{8}{3}, \ldots\) Text Transcription: frac23,frac43,2,frac83,ldots

In Exercises 19–22, match the sequence with the correct expression for its nth term. [The nth terms are labeled (a), (b), (c), and (d).] (a) \(a_{n}=\frac{2}{3} n\) (b) \(a_{n}=2-\frac{4}{n}\) (c) \(a_{n}=16(-0.5)^{n-1}\) (d) \(a_{n}=\frac{2 n}{n+1}\) \(1, \frac{4}{3}, \frac{3}{2}, \frac{8}{5}, \ldots\) Text Transcription: 1,frac43,frac32,frac85,ldots

In Exercises 23–28, write the next two apparent terms of the sequence. Describe the pattern you used to find these terms. \(2,5,8,11, \ldots\) Text Transcription: 2,5,8,11,ldots

In Exercises 23–28, write the next two apparent terms of the sequence. Describe the pattern you used to find these terms. \(\frac{7}{2}, 4, \frac{9}{2}, 5, \ldots\) Text Transcription: frac72,4,frac92,5,ldots

In Exercises 23–28, write the next two apparent terms of the sequence. Describe the pattern you used to find these terms. \(5,10,20,40, \ldots\) Text Transcription: 5,10,20,40,ldots

In Exercises 23–28, write the next two apparent terms of the sequence. Describe the pattern you used to find these terms. \(1,-\frac{1}{2}, \frac{1}{4},-\frac{1}{8}, \ldots\) Text Transcription: 1,-frac12,frac14,-frac18,ldots

In Exercises 23–28, write the next two apparent terms of the sequence. Describe the pattern you used to find these terms. \(3,-\frac{3}{2}, \frac{3}{4},-\frac{3}{8}, \ldots\) Text Transcription: 3,-frac32,frac34,-frac38,ldots

In Exercises 23–28, write the next two apparent terms of the sequence. Describe the pattern you used to find these terms. \(1,-\frac{3}{2}, \frac{9}{4},-\frac{27}{8}, \ldots\) Text Transcription: 1,-frac32,frac94,-frac278,ldots

In Exercises 29–34, simplify the ratio of factorials. \(\frac{11 !}{8 !}\) Text Transcription: frac11!8!

In Exercises 29–34, simplify the ratio of factorials. \(\frac{25 !}{20 !}\) Text Transcription: frac25!20!

In Exercises 29–34, simplify the ratio of factorials. \(\frac{(n+1) !}{n !}\) Text Transcription: frac(n+1)!n!

In Exercises 29–34, simplify the ratio of factorials. \(\frac{(n+2) !}{n !}\) Text Transcription: frac(n+2)!n!

In Exercises 29–34, simplify the ratio of factorials. \(\frac{(2 n-1) !}{(2 n+1) !}\) Text Transcription: frac(2n-1)!(2n+1)!

In Exercises 29–34, simplify the ratio of factorials. \(\frac{(2 n+2) !}{(2 n) !}\) Text Transcription: frac(2n+2)!(2n)!

In Exercises 35–40, find the limit (if possible) of the sequence. \(a_{n}=\frac{5 n^{2}}{n^{2}+2}\) Text Transcription: a_n=frac5n^2n^2+2

In Exercises 35–40, find the limit (if possible) of the sequence. \(a_{n}=5-\frac{1}{n^{2}}\) Text Transcription: a_n=5-frac1n^2

In Exercises 35–40, find the limit (if possible) of the sequence. \(a_{n}=\frac{2 n}{\sqrt{n^{2}+1}}\) Text Transcription: a_n=frac2nsqrtn^2+1

In Exercises 35–40, find the limit (if possible) of the sequence. \(a_{n}=\frac{5 n}{\sqrt{n^{2}+4}}\) Text Transcription: a_n=frac5nsqrtn^2+4

In Exercises 35–40, find the limit (if possible) of the sequence. \(a_{n}=\sin \frac{1}{n}\) Text Transcription: a_n=sinfrac1n

In Exercises 35–40, find the limit (if possible) of the sequence. \(a_{n}=\cos \frac{2}{n}\) Text Transcription: a_n=cosfrac2n

In Exercises 41–44, use a graphing utility to graph the first 10 terms of the sequence. Use the graph to make an inference about the convergence or divergence of the sequence. Verify your inference analytically and, if the sequence converges, find its limit. \(a_{n}=\frac{n+1}{n}\) Text Transcription: a_n=fracn+1n

In Exercises 41–44, use a graphing utility to graph the first 10 terms of the sequence. Use the graph to make an inference about the convergence or divergence of the sequence. Verify your inference analytically and, if the sequence converges, find its limit. \(a_{n}=\frac{1}{n^{3 / 2}}\) Text Transcription: a_n=frac1n^3/2

In Exercises 41–44, use a graphing utility to graph the first 10 terms of the sequence. Use the graph to make an inference about the convergence or divergence of the sequence. Verify your inference analytically and, if the sequence converges, find its limit. \(a_{n}=\cos \frac{n \pi}{2}\) Text Transcription: a_n=cosfracnpi2

In Exercises 41–44, use a graphing utility to graph the first 10 terms of the sequence. Use the graph to make an inference about the convergence or divergence of the sequence. Verify your inference analytically and, if the sequence converges, find its limit. \(a_{n}=3-\frac{1}{2^{n}}\) Text Transcription: a_n=3-frac12^n

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(\a_{n}=(0.3)^{n}-1) Text Transcription: a_n=(0.3)^n-1

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=4-\frac{3}{n}\) Text Transcription: a_n=4-frac3n

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\frac{5}{n+2}\) Text Transcription: a_n=frac5n+2

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\frac{2}{n !}\) Text Transcription: a_n=frac2n!

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=(-1)^{n}\left(\frac{n}{n+1}\right)\) Text Transcription: a_n=(-1)^n(fracnn+1)

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=1+(-1)^{n}\) Text Transcription: a_n=1+(-1)^n

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\frac{3 n^{2}-n+4}{2 n^{2}+1}\) Text Transcription: a_n=frac3n^2-n+42n^2+1

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\frac{\sqrt[3]{n}}{\sqrt[3]{n}+1}\) Text Transcription: a_n=fracsqrt[3]nsqrt[3]n+1

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\frac{1 \cdot 3 \cdot 5 \cdots \cdots(2 n-1)}{(2 n)^{n}}\) Text Transcription: a_n=frac1cdot3cdot5cdotscdots(2n-1)(2n)^n

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\frac{1 \cdot 3 \cdot 5 \cdots \cdot(2 n-1)}{n !}\) Text Transcription: a_n=frac1cdot3 cdot5cdotscdot(2 n-1)n!

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\frac{1+(-1)^{n}}{n}\) Text Transcription: a_n=frac1+(-1)^nn

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\frac{1+(-1)^{n}}{n^{2}}\) Text Transcription: a_n=frac1+(-1)^nn^2

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\frac{\ln \left(n^{3}\right)}{2 n}\) Text Transcription: a_n=fraclnleft(n^3)2n

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\frac{\ln \sqrt{n}}{n}\) Text Transcription: a_n=fraclnsqrtnn

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\frac{3^{n}}{4^{n}}\) Text Transcription: a_n=frac3^n4^n

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=(0.5)^{n}\) Text Transcription: a_n=(0.5)^n

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\frac{(n+1) !}{n !}\) Text Transcription: a_n=frac(n+1)!n!

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\frac{(n-2) !}{n !}\) Text Transcription: a_n=frac(n-2)!n!

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\frac{n-1}{n}-\frac{n}{n-1}, \quad n \geq 2\) Text Transcription: a_n=fracn-1n-fracnn-1,ngeq2

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\frac{n^{2}}{2 n+1}-\frac{n^{2}}{2 n-1}\) Text Transcription: a_n=fracn^22n+1-fracn^22n-1

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\frac{n^{p}}{e^{n}}, \quad p>0\) Text Transcription: a_n=fracn^pe^n,p>0

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=n \sin \frac{1}{n}\) Text Transcription: a_n=nsinfrac1n

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=2^{1 / n}\) Text Transcription: a_n=2^1n

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. a_{n}=-3^{-n} Text Transcription: a_n=-3^-n

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\left(1+\frac{k}{n}\right)^{n}\) Text Transcription: a_n=(1+frackn)^n

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\left(1+\frac{1}{n^{2}}\right)^{n}\) Text Transcription: a_n=(1+frac1n^2)^n

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\frac{\sin n}{n}\) Text Transcription: a_n=fracsinnn

In Exercises 45–72, determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\frac{\cos \pi n}{n^{2}}\) Text Transcription: a_n=fraccospinn^2

In Exercises 73–86, write an expression for the nth term of the sequence. (There is more than one correct answer.) \(1,4,7,10, \ldots\) Text Transcription: 1,4,7,10,ldots

In Exercises 73–86, write an expression for the nth term of the sequence. (There is more than one correct answer.) \(3,7,11,15, \ldots\) Text Transcription: 3,7,11,15,ldots

In Exercises 73–86, write an expression for the nth term of the sequence. (There is more than one correct answer.) \(-1,2,7,14,23, \ldots\) Text Transcription: -1,2,7,14,23,ldots

In Exercises 73–86, write an expression for the nth term of the sequence. (There is more than one correct answer.) \(1,-\frac{1}{4}, \frac{1}{9},-\frac{1}{16}, \ldots\) Text Transcription: 1,-frac14,frac19,-frac116,ldots

In Exercises 73–86, write an expression for the nth term of the sequence. (There is more than one correct answer.) \(\frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \ldots\) Text Transcription: frac23,frac34,frac45,frac56,ldots

In Exercises 73–86, write an expression for the nth term of the sequence. (There is more than one correct answer.) \(2,-1, \frac{1}{2},-\frac{1}{4}, \frac{1}{8}, \ldots\) Text Transcription: 2,-1,frac12,-frac14,frac18,ldots

In Exercises 73–86, write an expression for the nth term of the sequence. (There is more than one correct answer.) \(2,1+\frac{1}{2}, 1+\frac{1}{3}, 1+\frac{1}{4}, 1+\frac{1}{5}, \ldots\) Text Transcription: 2,1+frac12,1+frac13,1+frac14,1+frac15,ldots

In Exercises 73–86, write an expression for the nth term of the sequence. (There is more than one correct answer.) \(1+\frac{1}{2}, 1+\frac{3}{4}, 1+\frac{7}{8}, 1+\frac{15}{16}, 1+\frac{31}{32}, \ldots\) Text Transcription: 1+frac12,1+frac34,1+frac78,1+frac1516,1+frac3132,ldots

In Exercises 73–86, write an expression for the nth term of the sequence. (There is more than one correct answer.) \(\frac{1}{2 \cdot 3}, \frac{2}{3 \cdot 4}, \frac{3}{4 \cdot 5}, \frac{4}{5 \cdot 6}, \ldots\) Text Transcription: frac12cdot3,frac23cdot4,frac34cdot5,frac45cdot6,ldots

In Exercises 73–86, write an expression for the nth term of the sequence. (There is more than one correct answer.) \(1, \frac{1}{2}, \frac{1}{6}, \frac{1}{24}, \frac{1}{120}, \ldots\) Text Transcription: 1, frac12,frac16,frac124,frac1120,ldots

In Exercises 73–86, write an expression for the nth term of the sequence. (There is more than one correct answer.) \(1,-\frac{1}{1 \cdot 3}, \frac{1}{1 \cdot 3 \cdot 5},-\frac{1}{1 \cdot 3 \cdot 5 \cdot 7}, \ldots\) Text Transcription: 1,-frac11cdot3,frac11cdot3cdot,-frac11cdot3cdot5cdot7,ldots

In Exercises 73–86, write an expression for the nth term of the sequence. (There is more than one correct answer.) \(1, x, \frac{x^{2}}{2}, \frac{x^{3}}{6}, \frac{x^{4}}{24}, \frac{x^{5}}{120}, \ldots\) Text Transcription: 1,x,fracx^22,fracx^36,fracx^424,fracx^5120,ldots

In Exercises 73–86, write an expression for the nth term of the sequence. (There is more than one correct answer.) 2, 24, 720, 40,320, 3,628,800, . . .

In Exercises 73–86, write an expression for the nth term of the sequence. (There is more than one correct answer.) 1, 6, 120, 5040, 362,880, . . .

In Exercises 87–98, determine whether the sequence with the given th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results. \(a_{n}=4-\frac{1}{n}\) Text Transcription: a_n=4-frac1n

In Exercises 87–98, determine whether the sequence with the given th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results. \(a_{n}=\frac{3 n}{n+2}\) Text Transcription: a_n=frac3nn+2

In Exercises 87–98, determine whether the sequence with the given th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results. \(a_{n}=\frac{n}{2^{n+2}}\) Text Transcription: a_n=fracn2^n+2

In Exercises 87–98, determine whether the sequence with the given th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results. \(a_{n}=n e^{-n / 2}\) Text Transcription: a_n=ne^-n/2

In Exercises 87–98, determine whether the sequence with the given th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results. \(a_{n}=(-1)^{n}\left(\frac{1}{n}\right)\) Text Transcription: a_n=(-1)^n(frac1n)

In Exercises 87–98, determine whether the sequence with the given th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results. \(a_{n}=\left(-\frac{2}{3}\right)^{n}\) Text Transcription: a_n=(-frac23)^n

In Exercises 87–98, determine whether the sequence with the given th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results. \(a_{n}=\left(\frac{2}{3}\right)^{n}\) Text Transcription: a_n=(frac23)^n

In Exercises 87–98, determine whether the sequence with the given th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results. \(a_{n}=\left(\frac{3}{2}\right)^{n}\) Text Transcription: a_n=(frac32)^n

In Exercises 87–98, determine whether the sequence with the given th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results. \(a_{n}=\sin \frac{n \pi}{6}\) Text Transcription: a_n=sinfracnpi6

In Exercises 87–98, determine whether the sequence with the given th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results. \(a_{n}=\cos \frac{n \pi}{2}\) Text Transcription: a_n=cosfracnpi2

In Exercises 87–98, determine whether the sequence with the given th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results. \(a_{n}=\frac{\cos n}{n}\) Text Transcription: a_n=fraccosnn

In Exercises 87–98, determine whether the sequence with the given th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results. \(a_{n}=\frac{\sin \sqrt{n}}{n}\) Text Transcription: a_n=fracsinsqrtnn

In Exercises 99–102, (a) use Theorem 9.5 to show that the sequence with the given nth term converges and (b) use a graphing utility to graph the first 10 terms of the sequence and find its limit. \(a_{n}=5+\frac{1}{n}\) Text Transcription: a_n=5+frac1n

In Exercises 99–102, (a) use Theorem 9.5 to show that the sequence with the given nth term converges and (b) use a graphing utility to graph the first 10 terms of the sequence and find its limit. \(a_{n}=4-\frac{3}{n}\) Text Transcription: a_n=4-frac3n

In Exercises 99–102, (a) use Theorem 9.5 to show that the sequence with the given nth term converges and (b) use a graphing utility to graph the first 10 terms of the sequence and find its limit. \(a_{n}=\frac{1}{3}\left(1-\frac{1}{3^{n}}\right)\) Text Transcription: a_n=frac13(1-frac13^n)

In Exercises 99–102, (a) use Theorem 9.5 to show that the sequence with the given nth term converges and (b) use a graphing utility to graph the first 10 terms of the sequence and find its limit. \(a_{n}=4+\frac{1}{2^{n}}\) Text Transcription: a_n=4+frac12^n

Let \(\left\{a_{n}\right\}\) be an increasing sequence such that \(2 \leq a_{n} \leq 4\). Explain why \(\left\{a_{n}\right\}\) has a limit. What can you conclude about the limit? Text Transcription: {a_n} 2leq a_nleq4

Let \(\left\{a_{n}\right\}\) be a monotonic sequence such that \(a_{n} \leq 1\). Discuss the convergence of \(\left\{a_{n}\right\}\). If \(\left\{a_{n}\right\}\) converges, what can you conclude about its limit? Text Transcription: {a_n} a_nleq 1

Compound Interest Consider the sequence \(\left\{A_{n}\right\}\) whose th term is given by \(A_{n}=P\left(1+\frac{r}{12}\right)^{n}\) where is the principal, \(\left\{A_{n}\right\}\) is the account balance after months, and is the interest rate compounded annually. (a) Is \(\left\{A_{n}\right\}\) a convergent sequence? Explain. (b) Find the first 10 terms of the sequence if \(P=\$ 10,000\) and \(r=0.055\) Text Transcription: {A_n} A_n=P(1+fracr12)^n P=$10,000 r=0.055

Compound Interest A deposit of $100 is made at the beginning of each month in an account at an annual interest rate of 3% compounded monthly. The balance in the account after n months is \(A_{n}=100(401)\left(1.0025^{n}-1\right)\). (a) Compute the first six terms of the sequence \(\left\{A_{n}\right\}\). (b) Find the balance in the account after 5 years by computing the 60th term of the sequence. (c) Find the balance in the account after 20 years by computing the 240th term of the sequence. Text Transcription: {A_n} A_n=100(401)(1.0025^n-1)

Is it possible for a sequence to converge to two different numbers? If so, give an example. If not, explain why not.

In your own words, define each of the following. (a) Sequence (b) Convergence of a sequence (c) Monotonic sequence (d) Bounded sequence

The graphs of two sequences are shown in the figures. Which graph represents the sequence with alternating signs? Explain.

Give an example of a sequence satisfying the condition or explain why no such sequence exists. (Examples are not unique.) (a) A monotonically increasing sequence that converges to 10 (b) A monotonically increasing bounded sequence that does not converge (c) A sequence that converges to \(\frac{3}{4}\) (d) An unbounded sequence that converges to 100 Text Transcription: frac34

Government Expenditures A government program that currently costs taxpayers $4.5 billion per year is cut back by 20 percent per year. (a) Write an expression for the amount budgeted for this program after years. (b) Compute the budgets for the first 4 years. (c) Determine the convergence or divergence of the sequence of reduced budgets. If the sequence converges, find its limit.

Inflation If the rate of inflation is \(4 \frac{1}{2} \%\) per year and the average price of a car is currently $25,000, the average price after years is \(P_{n}=\$ 25,000(1.045)^{n}\) Text Transcription: 4frac12% P_n=$25,000(1.045)^n

Modeling Data The federal debts \(a_{n}\) (in billions of dollars) of the United States from 2002 through 2006 are shown in the table, where represents the year, with n = 2 corresponding to 2002. (Source: U.S. Office of Management and Budget) (a) Use the regression capabilities of a graphing utility to find a model of the form \(a_{n}=b n^{2}+c n+d, \quad n=2,3,4,5,6\) for the data. Use the graphing utility to plot the points and graph the model. (b) Use the model to predict the amount of the federal debt in the year 2012. Text Transcription: a_n a_n=b n^2+c n+d,n=2,3,4,5,6

Modeling Data The per capita personal incomes in the United States from 1996 through 2006 are given below as ordered pairs of the form \(\left(n, a_{n}\right)\), where represents the year, with corresponding to 1996. (Source: U.S. Bureau of Economic Analysis) (6, 24,176), (7, 25,334), (8, 26,880), (9, 27,933), (10, 29,855), (11, 30,572), (12, 30,805), (13, 31,469), (14, 33,102), (15, 34,493), (16, 36,313) (a) Use the regression capabilities of a graphing utility to find a model of the form \(a_{n}=b n+c, \quad n=6,7, \ldots, 16\) for the data. Graphically compare the points and the model. (b) Use the model to predict per capita personal income in the year 2012. Text Transcription: (n,a_n) a_n=bn+c,n=6,7,ldots,16

Comparing Exponential and Factorial Growth Consider the sequence \(a_{n}=10^{n} / n !\). (a) Find two consecutive terms that are equal in magnitude. (b) Are the terms following those found in part (a) increasing or decreasing? (c) In Section 8.7, Exercises 73–78, it was shown that for “large” values of the independent variable an exponential function increases more rapidly than a polynomial function. From the result in part (b), what inference can you make about the rate of growth of an exponential function versus a factorial function for “large” integer values of n? Text Transcription: a_n=10^n/n!

Compute the first six terms of the sequence \(\left\{a_{n}\right\}=\left\{\left(1+\frac{1}{n}\right)^{n}\right\}\) If the sequence converges, find its limit. Text Transcription: a_n=(1+frac1n)^n

Compute the first six terms of the sequence \(\left\{a_{n}\right\}=\{\sqrt[n]{n}\}\). If the sequence converges, find its limit. Text Transcription: a_n=sqrt[n]n

Prove that if \(\left\{s_{n}\right\}\) converges to and \(L>0\), then there exists a number N such that \(s_{n}>0\) for \(n>N\). Text Transcription: {s_n} L>0 s_n>0 n>N

True or False? In Exercises 119–124, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(\left\{a_{n}\right\}\) converges to 3 and \(\left\{b_{n}\right\}\) converges to 2, then \(\left\{a_{n}+b_{n}\right\}\) converges to 5. Text Transcription: {a_n} {b_n} {a_n}+b_n

True or False? In Exercises 119–124, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(\left\{a_{n}\right\}\) converges, then lim \(\left(a_{n}-a_{n+1}\right)=0\). Text Transcription: {a_n} (a_n-a_n+1)=0

True or False? In Exercises 119–124, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(n>1\), then Text Transcription: n>1

True or False? In Exercises 119–124, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(\left\{a_{n}\right\}\) converges, then \(\left\{a_{n} / n\right\}\) converges to 0. Text Transcription: {a_n} {a_n}/n

True or False? In Exercises 119–124, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(\left\{a_{n}\right\}\) converges to 0 and \(\left\{b_{n}\right\}\) is bounded, then \(\left\{a_{n}+b_{n}\right\}\) diverges. Text Transcription: {a_n} {b_n} {a_n}+b_n

True or False? In Exercises 119–124, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(\left\{a_{n}\right\}\) diverges and \(\left\{b_{n}\right\}\) diverges, then \(\left\{a_{n}+b_{n}\right\}\) diverges. Text Transcription: {a_n} {b_n} {a_n+b_n}

Fibonacci Sequence In a study of the progeny of rabbits, Fibonacci (ca. 1170–ca. 1240) encountered the sequence now bearing his name. The sequence is defined recursively as \(a_{n+2}=a_{n}+a_{n+1}\), where \(a_{1}=1\) and \(a_{2}=1\). (a) Write the first 12 terms of the sequence. (b) Write the first 10 terms of the sequence defined by \(b_{n}=\frac{a_{n+1}}{a_{n}}, \quad n \geq 1\). (c) Using the definition in part (b), show that \(b_{n}=1+\frac{1}{b_{n-1}}\). (d) The golden ratio can be defined by \(\lim _{n \rightarrow \infty} b_{n}=\rho\). Show that \(\rho=1+1 / \rho\) and solve this equation for p. Text Transcription: a_n+2=a_n+a_n+1 a_1=1 a_2=1 b_n=fraca_n+1a_n,ngeq1 b_n=1+frac1b_n-1 lim _nrightarrowinftyb_n=rho rho=1+1/rho

Conjecture Let \(x_{0}=1\) and consider the sequence \(x_{n}\) given by the formula \(x_{n}=\frac{1}{2} x_{n-1}+\frac{1}{x_{n-1}}, \quad n=1,2, \ldots\) Use a graphing utility to compute the first 10 terms of the sequence and make a conjecture about the limit of the sequence. Text Transcription: x_0=1 x_n x_n=frac12x_n-1+frac1x_n-1,n=1,2,ldots

Consider the sequence \(\sqrt{2}, \sqrt{2+\sqrt{2}}, \sqrt{2+\sqrt{2+\sqrt{2}}}, \ldots\) (a) Compute the first five terms of this sequence. (b) Write a recursion formula for \(a_{n}\), for \(n \geq 2\). (c) Find \(\lim _{n \rightarrow \infty} a_{n}\). Text Transcription: sqrt2,sqrt2+sqrt2,sqrt2+sqrt2+sqrt2,ldots a_n ngeq2 lim_ninftya_n

Consider the sequence \(\sqrt{6}, \sqrt{6+\sqrt{6}}, \sqrt{6+\sqrt{6+\sqrt{6}}}, \ldots\) (a) Compute the first five terms of this sequence. (b) Write a recursion formula for \(a_{n}\), for \(n \geq 2\). (c) Find \(\lim _{n \rightarrow \infty} a_{n}\). Text Transcription: sqrt6,sqrt6+sqrt6,sqrt6+sqrt6+sqrt6,ldots a_n ngeq2 lim_ninftya_n

Consider the sequence \(\left\{a_{n}\right\}\) where \(a_{1}=\sqrt{k}, a_{n+1}=\sqrt{k+a_{n}}\) (a) Show that \(\left\{a_{n}\right\}\) is increasing and bounded. (b) Prove that \(\lim _{n \rightarrow \infty} a_{n}\) exists. (c) Find \(\lim _{n \rightarrow \infty} a_{n}\). Text Transcription: {a_n} a_1=sqrtk,a_n+1=sqrtk+a_n lim_ninftya_n

Arithmetic-Geometric Mean Let \(a_{0}>b_{0}>0\). Let \(a_{1}\) be the arithmetic mean of \(a_{0}\) and \(b_{0}\) and let \(b_{1}\) be the geometric mean of \(a_{0}\) and \(b_{0}\). \(a_{1}=\frac{a_{0}+b_{0}}{2}\) Arithmetic Mean \(b_{1}=\sqrt{a_{0} b_{0}}\) Geometric Mean Now define the sequences \(\left\{a_{n}\right\}\) and \(\left\{b_{n}\right\}\) as follows. \(a_{n}=\frac{a_{n-1}+b_{n-1}}{2} \quad b_{n}=\sqrt{a_{n-1} b_{n-1}}\) (a) Let \(a_{0}=10\) and \(b_{0}=3\). Write out the first five terms of \(\left\{a_{n}\right\}\) and \(\left\{b_{n}\right\}\). What do you notice? (b) Use induction to show that \(a_{n}>a_{n+1}>b_{n+1}>b_{n}\), for \(a_{0}>b_{0}>0\). (c) Explain why \(\left\{a_{n}\right\}\) and \(\left\{b_{n}\right\}\) are both convergent. (d) Show that \(\lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} b_{n}\). Text Transcription: a_0>b_0>0 a_1 a_0 b_0 b_1) a_1=fraca_0+b_02 b_1=sqrta_0b_0 a_n b_n a_n=fraca_n-1+b_n-12b_n=sqrta_n-1b_n-1 a_0=10 b_0=3 a_n>a_n+1>b_n+1>b_n a_0>b_0>0 lim_ninftya_n=lim _ninftyb_n

(a) Let \(f(x)=\sin x\) and \(a_{n}=n \sin 1 / n\). Show that \(\lim _{n \rightarrow \infty} a_{n}=f^{\prime}(0)=1\). (b) Let f(x) be differentiable on the interval [0, 1] and f(0) = 0. Consider the sequence \(\left\{a_{n}\right\}\), where \(a_{n}=n f(1 / n)\). Show that \(\lim _{n \rightarrow \infty} a_{n}=f^{\prime}(0)\). Text Transcription: f(x)=sinx a_n=nsin1/n lim _ninftya_n=f^prime(0)=1 a_n a_n=nf(1/n) lim_ninftya_n=f^prime(0)

Consider the sequence \(\left\{a_{n}\right\}=\left\{n r^{n}\right\}\). Decide whether \(\left\{a_{n}\right\}\) converges for each value of r. (a) \(r=\frac{1}{2}\) (b) \(r=1\) (c) \(r=\frac{3}{2}\) (d) For what values of r does the sequence \(\left\{n r^{n}\right\}\) converge? Text Transcription: {a_n}={nr^n} {a_n} r=frac12 r=1 r=frac32 nr^n

(a) Show that \(\int_{1}^{n} \ln x d x<\ln (n !)\) for \(n \geq 2\). (b) Draw a graph similar to the one above that shows \(\ln (n !)<\int_{1}^{n+1} \ln x d x\). (c) Use the results of parts (a) and (b) to show that \(\frac{n^{n}}{e^{n-1}}<n !<\frac{(n+1)^{n+1}}{e^{n}}\), for \(n>1\). (d) Use the Squeeze Theorem for Sequences and the result of part (c) to show that \(\lim _{n \rightarrow \infty}(\sqrt[n]{n !} / n)=1 / e\). (e) Test the result of part (d) for n = 20, 50, and 100. Text Transcription: int_1^nlnxdx<In(n!) ln(n!)<int_1^n+1lnxdx fracn^ne^n-1<n!<frac(n+1)^n+1e^n n>1 lim_ninfty(sqrt[n]n!/n)=1/e

Consider the sequence \(\left\{a_{n}\right\}=\left\{\frac{1}{n} \sum_{k=1}^{n} \frac{1}{1+(k / n)}\right\}\). (a) Write the first five terms of \(\left\{a_{n}\right\}\). (b) Show that \(\lim _{n \rightarrow \infty} a_{n}=\ln 2\) by interpreting \(a_{n}\) as a Riemann sum of a definite integral. Text Transcription: {a_n}={\frac1nsum_k=1}^nfrac11+(k/n)} {a_n} lim_ninftya_n=ln2 a_n

Prove, using the definition of the limit of a sequence, that \(\lim _{n \rightarrow \infty} \frac{1}{n^{3}}=0\). Text Transcription: lim _ninftyfrac1n^3=0

Prove, using the definition of the limit of a sequence, that \(\lim _{n \rightarrow \infty} r^{n}=0\) for \(-1<r<1\). Text Transcription: lim_ninftyr^n=0 -1<r<1

Find a divergent sequence \(\left\{a_{n}\right\}\) such that \(\left\{a_{2 n}\right\}\) converges. Text Transcription: {a_n} {a_2n}

Show that the converse of Theorem 9.1 is not true. Hint: Find a function f(x) such that \(f(n)=a_{n}\) converges but \(\lim _{x \rightarrow \infty} f(x)\) does not exist.] Text Transcription: f(n)=a_n lim_xinftyf(x)

Prove Theorem 9.5 for a nonincreasing sequence.

Let \(\left\{x_{n}\right\}, n \geq 0\), be a sequence of nonzero real numbers such that \(x_{n}^{2}-x_{n-1} x_{n+1}=1) for \(n=1,2,3, \ldots\). Prove that there exists a real number a such that \(x_{n+1}=a x_{n}-x_{n-1}\), for all \(n \geq 1\). Text Transcription: x_n,ngeq0 x_n^2-x_n-1x_n+1=1 n=1,2,3,ldots x_n+1=ax_n-x_n-1 ngeq1

Let \(T_{0}=2, T_{1}=3, T_{2}=6\), and, for \(n \geq 3 \text {, }\) \(T_{n}=(n+4) T_{n-1}-4 n T_{n-2}+(4 n-8) T_{n-3}\). The first 10 terms of the sequence are 2, 3, 6, 14, 40, 152, 784, 5168, 40,576, 363,392. Find, with proof, a formula for \(T_{n}\) of the form \(T_{n}=A_{n}+B_{n}\), where \(\left\{A_{n}\right\}\) and \(\left\{B_{n}\right\}\) are well-known sequences. Text Transcription: T_0=2,T_1=3,T_2=6 ngeq3text T_n=(n+4)T_n-1-4nT_n-2+(4n-8)T_n-3 T_n T_n=A_n+B_n A_n B_n