Solved: In Exercises 67 and 68, (a) verify that the series

Listing the Terms of a Sequence In Exercises 1-4,write the first five terms of the sequence. \(a_{n}=5^{n}\) Text Transcription: a_n=5^n

Listing the Terms of a Sequence In Exercises 1-4,write the first five terms of the sequence. \(a_{n}=\frac{3^{n}}{n !}\) Text Transcription: a_n=3^n/n!

Listing the Terms of a Sequence In Exercises 1-4,write the first five terms of the sequence. \(a_{n}=\left(-\frac{1}{4}\right)^{n}\) Text Transcription: a_n=(-1/4)^n

Listing the Terms of a Sequence In Exercises 1-4,write the first five terms of the sequence. \(a_{n}=\frac{2 n}{n+5}\) Text Transcription: a_n=2n/n+5

Matching In Exercises 5-8, match the sequence with its graph. [The graphs are labeled (a), (b), (c), and (d).] \(a_{n}=4+\frac{2}{n}\) Text Transcription: a_n=4+2/n

Matching In Exercises 5-8, match the sequence with its graph. [The graphs are labeled (a), (b), (c), and (d).] \(a_{n}=4-\frac{1}{2} n\) Text Transcription: a_n=4-1/2 n

Matching In Exercises 5-8, match the sequence with its graph. [The graphs are labeled (a), (b), (c), and (d).] \(a_{n}=10(0.3)^{n-1}\) Text Transcription: a_n=10(0.3)^n-1

Matching In Exercises 5-8, match the sequence with its graph. [The graphs are labeled (a), (b), (c), and (d).] \(a_{n}=6\left(-\frac{2}{3}\right)^{n-1}\) Text Transcription: a_n=6(-2/3)^n-1

Finding the Limit of a Sequence In Exercises 9 and 10, 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. \(a_{n}=\frac{5 n+2}{n}\) Text Transcription: a_n=5n+2/n

Finding the Limit of a Sequence In Exercises 9 and 10, 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. \(a_{n}=\sin \frac{n \pi}{2}\) Text Transcription: a_n=sin n pi/2

Determining Convergence or Divergence In Exercises 11-18,determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\left(\frac{2}{5}\right)^{n}+5\) Text Transcription: a_n=(2/5)^n + 5

Determining Convergence or Divergence In Exercises 11-18,determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=3-\frac{2}{n^{2}-1}\) Text Transcription: a_n=3 - 2/n^2 - 1

Determining Convergence or Divergence In Exercises 11-18,determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\frac{n^{3}+1}{n^{2}}\) Text Transcription: a_n=n^3 + 1/n^2

Determining Convergence or Divergence In Exercises 11-18,determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\frac{1}{\sqrt{n}}\) Text Transcription: a_n=1/sqrt n

Determining Convergence or Divergence In Exercises 11-18,determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\frac{n}{n^{2}+1}\) Text Transcription: a_n=n/n^2+1

Determining Convergence or Divergence In Exercises 11-18,determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\frac{n}{\ln n}\) Text Transcription: a_n=n/ln n

Determining Convergence or Divergence In Exercises 11-18,determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\sqrt{n+1}-\sqrt{n}\) Text Transcription: a_n=sqrt n+1 - sqrt n

Determining Convergence or Divergence In Exercises 11-18,determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. \(a_{n}=\frac{\sin \sqrt{n}}{\sqrt{n}}\) Text Transcription: a_n=sin sqrt n/sqrt n

Finding the nth Term of a Sequence In Exercises 19-22, write an expression for the nth term of the sequence. (There is more than one correct answer.) 3, 8, 13, 18, 23 , . . .

Finding the nth Term of a Sequence In Exercises 19-22, write an expression for the nth term of the sequence. (There is more than one correct answer.) -5, -2, 3, 10, 19, . . .

Finding the nth Term of a Sequence In Exercises 19-22, write an expression for the nth term of the sequence. (There is more than one correct answer.) \(\frac{1}{2}, \frac{1}{3}, \frac{1}{7}, \frac{1}{25}, \frac{1}{121}, \cdots\) Text Transcription: 1/2,1/3,1/7,1/25,1/121, . . .

Finding the nth Term of a Sequence In Exercises 19-22, write an expression for the nth term of the sequence. (There is more than one correct answer.) \(\frac{1}{2}, \frac{2}{5}, \frac{3}{10}, \frac{4}{17}, \cdots\) Text Transcription: 1/2,2/5,3/10,4,17, . . .

Compound Interest A deposit of $8000 is made in an account that earns 5% interest compounded quarterly. The balance in the account after quarters is \(A_{n}=8000\left(1+\frac{0.05}{4}\right)^{n}, \quad n=1,2,3, \ldots\) (a) Compute the first eight terms of the sequence {\(A_{n}\)}. (b) Find the balance in the account after 10 years by computing the 40th term of the sequence. Text Transcription: A_n=8000(1+0.05/4)^n, n=1,2,3,... A_n

Depreciation A company buys a machine for S175,000. During the next 5 years, the machine will depreciate at a rate of 30% per year. (That is, at the end of each year, the depreciated value will be 70% of what it was at the beginning of the year.) (a) Find a formula for the nth term of the sequence that gives the value V of the machine t full years after it was purchased. (b) Find the depreciated value of the machine at the end of 5 full years.

Finding Partial Sums In Exercises 25 and 26, find the sequence of partial sums \(S_{1}, S_{2}, S_{3}, S_{4}\), and \(S_{5}\). \(3+\frac{3}{2}+1+\frac{3}{4}+\frac{3}{5}+\cdots\) Text Transcription: S_1, S_2, S_3. S_4 S_5 3+3/2+1+3/4+3/5+...

Finding Partial Sums In Exercises 25 and 26, find the sequence of partial sums \(S_{1}, S_{2}, S_{3}, S_{4}\), and \(S_{5}\). \(-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}-\frac{1}{32}+\cdots\) Text Transcription: S_1, S_2, S_3. S_4 S_5 -1/2+1/4-1/8+1/16-1/32+...

Numerical, Graphical, and Analytic Analysis In Exercises 27-30,(a) use a graphing utility to find the indicated partial sum and complete the table,and (b) use a graphing utility to graph the first 10 terms of the sequence of partial sums. \(\sum_{n=1}^{\infty}\left(\frac{3}{2}\right)^{n-1}\) Text Transcription: sum_n=1^infinity (3/2)^n-1

Numerical, Graphical, and Analytic Analysis In Exercises 27-30,(a) use a graphing utility to find the indicated partial sum and complete the table,and (b) use a graphing utility to graph the first 10 terms of the sequence of partial sums. \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{2 n}\) Text Transcription: sum_n=1^infinity (-1)^n+1/2n

Numerical, Graphical, and Analytic Analysis In Exercises 27-30,(a) use a graphing utility to find the indicated partial sum and complete the table,and (b) use a graphing utility to graph the first 10 terms of the sequence of partial sums. \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{(2 n) !}\) Text Transcription: sum_n=1^infinity (-1)^n+1/(2n)!

Numerical, Graphical, and Analytic Analysis In Exercises 27-30,(a) use a graphing utility to find the indicated partial sum and complete the table,and (b) use a graphing utility to graph the first 10 terms of the sequence of partial sums. \(\sum_{n=1}^{\infty} \frac{1}{n(n+1)}\) Text Transcription: sum_n=1^infinity 1/n(n+1)

Finding the Sum of a Convergent Series In Exercises 31-34,find the sum of the convergent series. \(\sum_{n=0}^{\infty}\left(\frac{2}{5}\right)^{n}\) Text Transcription: sum_n=0^infinity (2/5)^n

Finding the Sum of a Convergent Series In Exercises 31-34,find the sum of the convergent series. \(\sum_{n=0}^{\infty} \frac{3^{n+2}}{7^{n}}\) Text Transcription: sum_n=0^infinity 3^n+2/7^n

Finding the Sum of a Convergent Series In Exercises 31-34,find the sum of the convergent series. \(\sum_{n=1}^{\infty}\left[(0.6)^{n}+(0.8)^{n}\right]\) Text Transcription: sum_n=1^infinity [(0.6)^n+(0.8)^n]

Finding the Sum of a Convergent Series In Exercises 31-34,find the sum of the convergent series. \(\sum_{n=0}^{\infty}\left[\left(\frac{2}{3}\right)^{n}-\frac{1}{(n+1)(n+2)}\right]\) Text Transcription: sum_n=0^infinity [(2/3)^n - 1/(n+1)(n+2)]

Using a Geometric Series In Exercises 35 and 36, (a) write the repeating decimal as a geometric series,and (b) write its sum as the ratio of two integers. \(0 . \overline{09}\) Text Transcription: 0. Overline 09

Using a Geometric Series In Exercises 35 and 36, (a) write the repeating decimal as a geometric series,and (b) write its sum as the ratio of two integers. \(0 . \overline{64}\) Text Transcription: 0. Overline 64

Using Geometric Series or the nth-Term Test In Exercises 37-40,use geometric series or the nth-Term Test to determine the convergence or divergence of the series. \(\sum_{n=0}^{\infty}(1.67)^{n}\) Text Transcription: sum_n=0^infinity (1.67)^n

Using Geometric Series or the nth-Term Test In Exercises 37-40,use geometric series or the nth-Term Test to determine the convergence or divergence of the series. \(\sum_{n=0}^{\infty}(0.36)^{n}\) Text Transcription: sum_n=0^infinity (0.36)^n

Using Geometric Series or the nth-Term Test In Exercises 37-40,use geometric series or the nth-Term Test to determine the convergence or divergence of the series. \(\sum_{n=2}^{\infty} \frac{(-1)^{n} n}{\ln n}\) Text Transcription: sum_n=2^infinity (-1)^n n/ln n

Using Geometric Series or the nth-Term Test In Exercises 37-40,use geometric series or the nth-Term Test to determine the convergence or divergence of the series. \(\sum_{n=0}^{\infty} \frac{2 n+1}{3 n+2}\) Text Transcription: sum_n=0^infinity 2n+1/3n+2

Distance A ball is dropped from a height of 8 meters. Each time it drops h meters, it rebounds 0.7h meters. Find the total distance traveled by the ball.

Compound Interest A deposit of $125 is made at the end of each month for 10 years in an account that pays 3.5% interest, compounded monthly. Determine the balance in the account at the end of 10 years. (Hint: Use the result of Section 9.2, Exercise 84.)

Using the Integral Test or a p-Series In Exercises 43-48,use the Integral Test or a p-series to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{2}{6 n+1}\) Text Transcription: sum_n=1^infinity 2/6n+1

Using the Integral Test or a p-Series In Exercises 43-48,use the Integral Test or a p-series to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{1}{\sqrt[4]{n^{3}}}\) Text Transcription: sum_n=1^infinity 1/4 sqrt n^3

Using the Integral Test or a p-Series In Exercises 43-48,use the Integral Test or a p-series to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{1}{n^{5 / 2}}\) Text Transcription: sum_n=1^infinity 1/n^5/2

Using the Integral Test or a p-Series In Exercises 43-48,use the Integral Test or a p-series to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{1}{5^{n}}\) Text Transcription: sum_n=1^infinity 1/5^n

Using the Integral Test or a p-Series In Exercises 43-48,use the Integral Test or a p-series to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty}\left(\frac{1}{n^{2}}-\frac{1}{n}\right)\) Text Transcription: sum_n=1^infinity (1/n^2-1/n)

Using the Integral Test or a p-Series In Exercises 43-48,use the Integral Test or a p-series to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{\ln n}{n^{4}}\) Text Transcription: sum_n=1^infinity ln n/n^4

Using the Direct Comparison Test or the Limit Comparison Test In Exercises 49-54,use the Direct Comparison Test or the Limit Comparison Test to determine the convergence or divergence of the series. \(\sum_{n=2}^{\infty} \frac{1}{\sqrt[3]{n}-1}\) Text Transcription: sum_n=2^infinity 1/3 sqrt n - 1

Using the Direct Comparison Test or the Limit Comparison Test In Exercises 49-54,use the Direct Comparison Test or the Limit Comparison Test to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{n}{\sqrt{n^{3}+3 n}}\) Text Transcription: sum_n=1^infinity n/sqrt n^3 + 3n

Using the Direct Comparison Test or the Limit Comparison Test In Exercises 49-54,use the Direct Comparison Test or the Limit Comparison Test to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{1}{\sqrt{n^{3}+2 n}}\) Text Transcription: sum_n=1^infinity 1/sqrt n^3+2n

Using the Direct Comparison Test or the Limit Comparison Test In Exercises 49-54,use the Direct Comparison Test or the Limit Comparison Test to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{n+1}{n(n+2)}\) Text Transcription: sum_n=1^infinity n+1/n(n+2)

Using the Direct Comparison Test or the Limit Comparison Test In Exercises 49-54,use the Direct Comparison Test or the Limit Comparison Test to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdot \cdot(2 n-1)}{2 \cdot 4 \cdot 6 \cdot \cdot \cdot(2 n)}\) Text Transcription: sum_n=1^infinity 1 times 3 times 5 … (2n-1)

Using the Direct Comparison Test or the Limit Comparison Test In Exercises 49-54,use the Direct Comparison Test or the Limit Comparison Test to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{1}{3^{n}-5}\) Text Transcription: sum_n=1^infinity 1/3^n-5

Using the Alternating Series Test In Exercises 55-60, use the Alternating Series Test,if applicable,to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{5}}\) Text Transcription: sum_n=1^infinity (-1)^n/n^5

Using the Alternating Series Test In Exercises 55-60, use the Alternating Series Test,if applicable,to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{(-1)^{n}(n+1)}{n^{2}+1}\) Text Transcription: sum_n=1^infinity (-1)^n (n+1)/n^2 + 1

Using the Alternating Series Test In Exercises 55-60, use the Alternating Series Test,if applicable,to determine the convergence or divergence of the series. \(\sum_{n=2}^{\infty} \frac{(-1)^{n} n}{n^{2}-3}\) Text Transcription: sum_n=2^infinity (-1)^n n/n^2 - 3

Using the Alternating Series Test In Exercises 55-60, use the Alternating Series Test,if applicable,to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{(-1)^{n} \sqrt{n}}{n+1}\) Text Transcription: sum_n=1^infinity (-1)^n sqrt n

Using the Alternating Series Test In Exercises 55-60, use the Alternating Series Test,if applicable,to determine the convergence or divergence of the series. \(\sum_{n=4}^{\infty} \frac{(-1)^{n} n}{n-3}\) Text Transcription: sum_n=4^infinity (-1)^n n/n - 3

Using the Alternating Series Test In Exercises 55-60, use the Alternating Series Test,if applicable,to determine the convergence or divergence of the series. \(\sum_{n=2}^{\infty} \frac{(-1)^{n} \ln n^{3}}{n}\) Text Transcription: sum_n=2^infinity (-1)^n ln n^3/n

Using the Ratio Test or the Root Test In Exercises 61-66,use the Ratio Test or the Root Test to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty}\left(\frac{3 n-1}{2 n+5}\right)^{n}\) Text Transcription: sum_n=1^infinity (3n-1/2n+5)^n

Using the Ratio Test or the Root Test In Exercises 61-66,use the Ratio Test or the Root Test to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty}\left(\frac{4 n}{7 n-1}\right)^{n}\) Text Transcription: sum_n=1^infinity (4n/7n-1)^n

Using the Ratio Test or the Root Test In Exercises 61-66,use the Ratio Test or the Root Test to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{n}{e^{n^{2}}}\) Text Transcription: sum_n=1^infinity n/e^n^2

Using the Ratio Test or the Root Test In Exercises 61-66,use the Ratio Test or the Root Test to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{n !}{e^{n}}\) Text Transcription: sum_n=1^infinity n!/e^n

Using the Ratio Test or the Root Test In Exercises 61-66,use the Ratio Test or the Root Test to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{2^{n}}{n^{3}}\) Text Transcription: sum_n=1^infinity 2^n/n^3

Using the Ratio Test or the Root Test In Exercises 61-66,use the Ratio Test or the Root Test to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdot \cdot(2 n-1)}{2 \cdot 5 \cdot 8 \cdot \cdots(3 n-1)}\) Text Transcription: sum_n=1^infinity 1 times 3 times 5 … (2n-1)/2 times 5 times 8 … (3n-1)

Numerical, Graphical, and Analytic Analysis In Exercises 67 and 68,(a) verify that the series converges,(b) use a graphing utility to find the indicated partial sum and complete the table,(c) use a graphing utility to graph the first 10 terms of the sequence of partial sums,and (d) use the table to estimate the sum of the series. \(\sum_{n=1}^{\infty} n\left(\frac{3}{5}\right)^{n}\) Text Transcription: sum_n=1^infinity n(3/5)^n

Numerical, Graphical, and Analytic Analysis In Exercises 67 and 68,(a) verify that the series converges,(b) use a graphing utility to find the indicated partial sum and complete the table,(c) use a graphing utility to graph the first 10 terms of the sequence of partial sums,and (d) use the table to estimate the sum of the series. \(\sum_{n=1}^{\infty} \frac{(-1)^{n-1} n}{n^{3}+5}\) Text Transcription: sum_n=1^infinity (-1)^n-1 n/n^3 + 5

Finding a Maclaurin Polynomial In Exercises 69 and 70, find the nth Maclaurin polynomial for the function. \(f(x)=e^{-2 x}, \quad n=3\) Text Transcription: f(x)=e^-2x, n=3

Finding a Maclaurin Polynomial In Exercises 69 and 70, find the nth Maclaurin polynomial for the function. \(f(x)=\cos \pi x, \quad n=4\) Text Transcription: f(x)=cos pi x, n=4

Finding a Taylor Polynomial In Exercises 71 and 72, find the third-degree Taylor polynomial centered at c. \(f(x)=e^{-3 x}, \quad c=0\) Text Transcription: f(x)=e^-3x, c=0

Finding a Taylor Polynomial In Exercises 71 and 72, find the third-degree Taylor polynomial centered at c. \(f(x)=\tan x, \quad c=-\frac{\pi}{4}\) Text Transcription: f(x)=tan x, c=-pi/4

Finding a Degree In Exercises 73 and 74, determine the degree of the Maclaurin polynomial required for the error in the approximation of the function at the indicated value of r to be less than 0.001. cos(0.75)

Finding a Degree In Exercises 73 and 74, determine the degree of the Maclaurin polynomial required for the error in the approximation of the function at the indicated value of r to be less than 0.001. \(e^{-0.25}\) Text Transcription: e^-0.25

Finding the Interval of Convergence In Exercises 75-80, find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) \(\sum_{n=0}^{\infty}\left(\frac{x}{10}\right)^{n}\) Text Transcription: sum_n=0^infinity (x/10)^n

Finding the Interval of Convergence In Exercises 75-80, find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) \(\sum_{n=0}^{\infty}(5 x)^{n}\) Text Transcription: sum_n=0^infinity (5x)^n

Finding the Interval of Convergence In Exercises 75-80, find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) \(\sum_{n=0}^{\infty} \frac{(-1)^{n}(x-2)^{n}}{(n+1)^{2}}\) Text Transcription: sum_n=0^infinity (-1)^n (x-2)^n/(n+1)^2

Finding the Interval of Convergence In Exercises 75-80, find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) \(\sum_{n=1}^{\infty} \frac{3^{n}(x-2)^{n}}{n}\) Text Transcription: sum_n=1^infinity 3^n(x-2)^n/n

Finding the Interval of Convergence In Exercises 75-80, find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) \(\sum_{n=0}^{\infty} n !(x-2)^{n}\) Text Transcription: sum_n=0^infinity n!(x-2)^n

Finding the Interval of Convergence In Exercises 75-80, find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) \(\sum_{n=0}^{\infty} \frac{(x-2)^{n}}{2^{n}}\) Text Transcription: sum_n=0^infinity (x-2)^n/2^n

Finding Intervals of Convergence In Exercises 81 and 82, find the intervals of convergence of (a) f(x), (b) f’(x), (c) f’’(x), and (d) \(\int f(x) d x\). Include a check for convergence at the endpoints of the interval. \(f(x)=\sum_{n=0}^{\infty}\left(\frac{x}{5}\right)^{n}\) Text Transcription: int f(x) dx f(x)=sum_n=0^infinity (x/5)^n

Finding Intervals of Convergence In Exercises 81 and 82, find the intervals of convergence of (a) f(x), (b) f’(x), (c) f’’(x), and (d) \(\int f(x) d x\). Include a check for convergence at the endpoints of the interval. \(f(x)=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x-4)^{n}}{n}\) Text Transcription: int f(x) dx f(x)=sum_n=1^infinity (-1)^n+1 (x-4)^n/n

Differential Equation In Exercises 83 and 84,show that the function represented by the power series is a solution of the differential equation. \(\begin{array}{l}y=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n}}{4^{n}(n !)^{2}} \\x^{2} y^{\prime \prime}+x y^{\prime}+x^{2} y=0\end{array}\) Text Transcription: y=sum_n=0^infinity (-1)^n x^2n/4^n(n!)^2 x^2y’’ + xy’ + x^2 y=0

Differential Equation In Exercises 83 and 84,show that the function represented by the power series is a solution of the differential equation. \(\begin{array}{l}y=\sum_{n=0}^{\infty} \frac{(-3)^{n} x^{2 n}}{2^{n} n !} \\y^{\prime \prime}+3 x y^{\prime}+3 y=0\end{array}\) Text Transcription: y=sum_n=0^infinity (-3)^n x^2n/2^n n! y’’ + 3xy’ + 3y=0

Finding a Geometric Power Series In Exercises 85 and 86, find a geometric power series,centered at 0,for the function. \(g(x)=\frac{2}{3-x}\) Text Transcription: g(x)=2/3-x

Finding a Geometric Power Series In Exercises 85 and 86, find a geometric power series,centered at 0,for the function. \(h(x)=\frac{3}{2+x}\) Text Transcription: h(x)=3/2+x

Finding a Power Series In Exercises 87 and 88,find a power series for the function,centered at and determine the interval of convergence. \(f(x)=\frac{6}{4-x}, \quad c=1\) Text Transcription: f(x)=6/4-x, c=1

Finding a Power Series In Exercises 87 and 88,find a power series for the function,centered at and determine the interval of convergence. \(f(x)=\frac{1}{3-2 x}, \quad c=0\) Text Transcription: f(x)=1/3-2x, c=0

Finding theSum of a Series In Exercises 89-94,find the sum of the convergent series by using a well-known function. Identify the function and explain how you obtained the sum. \(\sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{4^{n} n}\) Text Transcription: sum_n=1^infinity (-1)^n+1 1/4^n n

Finding theSum of a Series In Exercises 89-94,find the sum of the convergent series by using a well-known function. Identify the function and explain how you obtained the sum. \(\sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{5^{n} n}\) Text Transcription: sum_n=1^infinity (-1)^n+1 1/5^n n

Finding theSum of a Series In Exercises 89-94,find the sum of the convergent series by using a well-known function. Identify the function and explain how you obtained the sum. \(\sum_{n=0}^{\infty} \frac{1}{2^{n} n !}\) Text Transcription: sum_n=0^infinity 1/2^n n!

Finding theSum of a Series In Exercises 89-94,find the sum of the convergent series by using a well-known function. Identify the function and explain how you obtained the sum. \(\sum_{n=0}^{\infty} \frac{2^{n}}{3^{n} n !}\) Text Transcription: sum_n=0^infinity 2^n/3^n n!

Finding theSum of a Series In Exercises 89-94,find the sum of the convergent series by using a well-known function. Identify the function and explain how you obtained the sum. \(\sum_{n=0}^{\infty}(-1)^{n} \frac{2^{2 n}}{3^{2 n}(2 n) !}\) Text Transcription: sum_n=0^infinity (-1)^n 2^2n/3^2n (2n)!

Finding theSum of a Series In Exercises 89-94,find the sum of the convergent series by using a well-known function. Identify the function and explain how you obtained the sum. \(\sum_{n=0}^{\infty}(-1)^{n} \frac{1}{3^{2 n+1}(2 n+1) !}\) Text Transcription: sum_n=0^infinity (-1)^n 1/3^2n+1 (2n+1)!

Finding a Taylor Series In Exercises 95-102,use the definition of Taylor series to find the Taylor series,centered at c for the function. \(f(x)=\sin x, \quad c=\frac{3 \pi}{4}\) Text Transcription: f(x)=sin x, c=3 pi/4

Finding a Taylor Series In Exercises 95-102,use the definition of Taylor series to find the Taylor series,centered at c for the function. \(f(x)=\cos x, \quad c=-\frac{\pi}{4}\) Text Transcription: f(x)=cos x, c=-pi/4

Finding a Taylor Series In Exercises 95-102,use the definition of Taylor series to find the Taylor series,centered at c for the function. \(f(x)=3^{x}, \quad c=0\) Text Transcription: f(x)=3^x, c=0

Finding a Taylor Series In Exercises 95-102,use the definition of Taylor series to find the Taylor series,centered at c for the function. \(f(x)=\csc x, \quad c=\frac{\pi}{2}\) (first three terms) Text Transcription: f(x)=csc x, c=pi/2

Finding a Taylor Series In Exercises 95-102,use the definition of Taylor series to find the Taylor series,centered at c for the function. \(f(x)=\frac{1}{x}, \quad c=-1\) Text Transcription: f(x)=1/x, c=-1

Finding a Taylor Series In Exercises 95-102,use the definition of Taylor series to find the Taylor series,centered at c for the function. \(f(x)=\sqrt{x}, \quad c=4\) Text Transcription: f(x)=sqrt x, c=4

Finding a Taylor Series In Exercises 95-102,use the definition of Taylor series to find the Taylor series,centered at c for the function. \(g(x)=\sqrt[5]{1+x}, \quad c=0\) Text Transcription: g(x)=5 sqrt 1+x, c=0

Finding a Taylor Series In Exercises 95-102,use the definition of Taylor series to find the Taylor series,centered at c for the function. \(h(x)=\frac{1}{(1+x)^{3}}, \quad c=0\) Text Transcription: h(x)=1/(1+x)^3, c=0

Forming Maclaurin Series Determine the first four terms of the Maclaurin series for \(e^{2 x}\) (a) by using the definition of the Maclaurin series and the formula for the coefficient of the nth term, \(a_{n}=f^{(n)}(0) / n !\) (b) by replacing x by 2x in the series for \(e^{x\). (c) by multiplying the series for \(e^{x\) by itself, because \(e^{2 x}=e^{x} \cdot e^{x}\). Text Transcription: e^2x a_n=f^(n) (0) / n ! e^x e^2x = e^x times e^x

Forming Maclaurin Series Determine the first four terms of the Maclaurin series for sin 2x (a) by using the definition of the Maclaurin series and the formula for the coefficient of the nth term, \(a_{n}=f^{(n)}(0) / n !\) (b) by replacing x by 2x in the series for sin 2x. (c) by multiplying 2 by the series for sin x by the series for cos x, because sin 2x = 2 sin x cos x. Text Transcription: a_{n}=f^(n)(0) / n !

Finding a Maclaurin Series In Exercises 105-108,find the Maclaurin series for the function. Use the table of power series for elementary functions on page 670. \(f(x)=e^{6 x}\) Text Transcription: f(x)=e^6x

Finding a Maclaurin Series In Exercises 105-108,find the Maclaurin series for the function. Use the table of power series for elementary functions on page 670. f(x) = ln (x-1)

Finding a Maclaurin Series In Exercises 105-108,find the Maclaurin series for the function. Use the table of power series for elementary functions on page 670. f(x) = sin 2x

Finding a Maclaurin Series In Exercises 105-108,find the Maclaurin series for the function. Use the table of power series for elementary functions on page 670. f(x) = cos 3x

Finding a Limit In Exercises 109 and 110, use the series representation of the function f to find \(\lim _{x \rightarrow 0} f(x)\) (if it exists). \(f(x)=\frac{\arctan x}{\sqrt{x}}\) Text Transcription: lim_x rightarrow 0 f(x) f(x)=arctan x/sqrt x

Finding a Limit In Exercises 109 and 110, use the series representation of the function f to find \(\lim _{x \rightarrow 0} f(x)\) (if it exists). \(f(x)=\frac{\arcsin x}{x}\) Text Transcription: lim_x rightarrow 0 f(x) f(x)=arcsin x/x