?A certain ball has the property that each time it falls from a height h onto a hard

(a) What is the difference between a sequence and a series? (b) What is a convergent series? What is a divergent series?

Explain what it means to say that \(\sum_{n=1}^{\infty} a_{n}=5\). Equation Transcription: Text Transcription: sum_n=1^infinity a_n=5

Calculate the sum of the series \(\sum_{n=1}^{\infty} a_n\) whose partial sums are given. \(s_n=2-3(0.8)^n\)

Calculate the sum of the series \(\sum_{n=1}^{\infty} a_{n}\) whose partial sums are given. \(s_{n}=\frac{n^{2}-1}{4 n^{2}+1}\) Equation Transcription: Text Transcription: sum_n=1^infinity a_n s_n+ n^2 - 1/ 4n^2 + 1

Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent? \(\sum_{n=1}^{\infty} \frac{1}{n^{3}}\) Equation Transcription: Text Transcription: sum_n=1^infinity 1/n^3

Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent? \(\sum_{n=1}^{\infty} \frac{1}{\sqrt[3]{n}}\) Equation Transcription: Text Transcription: sum_n=1^infinity 1/cube root n

Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent? \(\sum_{n=1}^{\infty}\sin n\) Equation Transcription: Text Transcription: the sum from n =1 to infinity of sin n

Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent? \(\sum_{n=1}^{\infty}(-1)^{n} n\) Equation Transcription: Text Transcription: sum_n=1^infinity (-1)^n n

Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent? \(\sum_{n=1}^{\infty} \frac{1}{n^{4}+n^{2}}\) Equation Transcription: Text Transcription: sum_n=1^infinity 1/n^4+n^2

Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent? \(\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n !}\) Equation Transcription: Text Transcription: sum_n=1^infinity (-1)^n-1/n!

Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why. \(\sum_{n=1}^{\infty} \frac{6}{(-3)^{n}}\)

Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why. \(\sum_{n=1}^{\infty} \cos n\) Equation Transcription: Text Transcription: the sum from n =1 to infinity cos n

Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why. \(\sum_{n=1}^{\infty} \frac{n}{\sqrt{n^{2}+4}}\) Equation Transcription: Text Transcription: sum _n=1^infinity n/sqrt n^2 + 4

Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why. \(\sum_{n=1}^{\infty} \frac{7^{n+1}}{10^{n}}\)

Let \(a_{n}=\frac{2 n}{3 n+1}\). (a) Determine whether \(\left\{a_{n}\right\}\) is convergent. (b) Determine whether \(\sum_{n-1}^{\infty} a_{n}\) is convergent. Equation Transcription: Text Transcription: a_n=2n/3n+1 {an} sum_n-1^infinity a_n

(a) Explain the difference between \(\sum_{i=1}^{n} a_{i} \quad \text { and } \quad \sum_{j=1}^{n} a_{j}\) (b) Explain the difference between \(\sum_{i=1}^{n} a_{i} \quad \text { and } \quad \sum_{i=1}^{n} a_{j}\) Equation Transcription: Text Transcription: sum_-=1^n a_i sum_j=1^n a_j sum_i=1^n a_j

Determine whether the series is convergent or divergent by expressing \(S_{n}\) as a telescoping sum (as in Example 2). If it is convergent, find its sum. \(\sum_{n=1}^{\infty}\left(\frac{1}{n+2}-\frac{1}{n}\right)\) Equation Transcription: Text Transcription: s_n sum _n=1^infinity (1/n+2-1/n)

Determine whether the series is convergent or divergent by expressing \(S_{n}\) as a telescoping sum (as in Example 2). If it is convergent, find its sum. \(\sum_{n=4}^{\infty}\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)\) Equation Transcription: Text Transcription: s_n sum _n=4^infinity (1/square root n-1/square root n+1

Determine whether the series is convergent or divergent by expressing \(S_{n}\) as a telescoping sum (as in Example 2). If it is convergent, find its sum. \(\sum_{n=1}^{\infty} \frac{3}{n(n+3)}\) Equation Transcription: Text Transcription: s_n sum_n=1^infinity 3/n(n+3)

Determine whether the series is convergent or divergent by expressing \(S_{n}\) as a telescoping sum (as in Example 2). If it is convergent, find its sum. \(\sum_{n=1}^{\infty} \ln \frac{n}{n+1}\) Equation Transcription: Text Transcription: s_n sum_n=1^infinity ln n/n+1

Determine whether the series is convergent or divergent by expressing \(s_{n}\) as a telescoping sum (as in Example 2). If it is convergent, find its sum. \(\sum_{n=1}^{\infty}\left(e^{1 / n}-e^{1 /(n+1)}\right)\)

Determine whether the series is convergent or divergent by expressing \(S_{n}\) as a telescoping sum (as in Example 2). If it is convergent, find its sum. \(\sum_{n=2}^{\infty} \frac{1}{n^{3}-n}\) Equation Transcription: Text Transcription: s_n sum_n=2^infinity1/n^3-n

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum \(3-4+\frac{16}{3}-\frac{64}{9}+\cdots\)

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum \(4+3+\frac{9}{4}+\frac{27}{16}+\cdots\) ________________ Equation Transcription: Text Transcription: 4+3+9/4+27/16+dots

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum \(10-2+0.4-0.08+\cdots\) Equation Transcription: Text Transcription: 10-2+0.4-0.08+dots

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum \(2+0.5+0.125+0.03125+\cdots\) Equation Transcription: Text Transcription: 2+0.5+0.125+0.03125+dots

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum \(\sum_{n=1}^{\infty} 12(0.73)^{n-1}\)

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum \(\sum_{n=1}^{\infty} \frac{5}{\pi^{n}}\) Equation Transcription: Text Transcription: sum_n=1^infinity 5/pi^n

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum \(\sum_{n=1}^{\infty} \frac{(-3)^{n-1}}{4^n}\)

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum \(\sum_{n=0}^{\infty} \frac{3^{n+1}}{(-2)^{n}}\) Equation Transcription: Text Transcription: sum_n=0^infinity 3^n+1/(-2)^n

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum \(\sum_{n=1}^{\infty} \frac{e^{2 n}}{6^{n-1}}\) Equation Transcription: Text Transcription: sum_n=1^infinity e^2n/6^n-1

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum \(\sum_{n=1}^{\infty} \frac{6 \cdot 2^{2 n-1}}{3^{n}}\) Equation Transcription: Text Transcription: sum_n=1^infinity 6 dot 2^2n-?^n

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \(\frac{1}{3}+\frac{1}{6}+\frac{1}{9}+\frac{1}{12}+\frac{1}{15}+\cdots\) Equation Transcription: Text Transcription: 1/3+1/6+1/9+1/12+1/15+dots

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \(\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\frac{4}{5}+\frac{5}{6}+\frac{6}{7}+\cdots\)

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \(\frac{2}{5}+\frac{4}{25}+\frac{8}{125}+\frac{16}{625}+\frac{32}{3125}+\cdots\) Equation Transcription: Text Transcription: 2/5+4/25+8/125+16/625+32/3125+dots

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \(\frac{1}{3}+\frac{2}{9}+\frac{1}{27}+\frac{2}{81}+\frac{1}{243}+\frac{2}{729}+\cdots\) Equation Transcription: Text Transcription: 1/3+2/9+1/27+2/81+1/243+2/729+dots

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \(\sum_{n=1}^{\infty} \frac{2+n}{1-2 n}\)

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \(\sum_{k=1}^{\infty} \frac{k^{2}}{k^{2}-2 k+5}\) Equation Transcription: Text Transcription: sum _n=1^infinity k^2/k^2 - 2k + 5

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \(\sum_{n=1}^{\infty} 3^{n+1} 4^{-n}\)

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \(\sum_{n=1}^{\infty}\left[(-0.2)^{n}+(0.6)^{n-1}\right]\) Equation Transcription: Text Transcription: sum_n=1^infinity [(-0.2)^n+(0.6)^n-1]

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \(\sum_{n=1}^{\infty} \frac{1}{4+e^{-n}}\) Equation Transcription: Text Transcription: sum_n=1^infinity 1/4+e^-n

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \(\sum_{n=1}^{\infty} \frac{2^{n}+4^{n}}{e^{n}}\) Equation Transcription: Text Transcription: sum _n=1^infinity 2^n + 4^n/e^n

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \(\sum_{k=1}^{\infty}(\sin 100)^{k}\) Equation Transcription: Text Transcription: the sum from k =1 to infinity (sin 100)^k

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \(\sum_{n=1}^{\infty} \frac{1}{1+\left(\frac{2}{3}\right)^{n}}\) Equation Transcription: Text Transcription: sum _n=1^infinity 1/1 + (?)^n

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \(\sum_{n=1}^{\infty} \ln \left(\frac{n^{2}+1}{2 n^{2}+1}\right)\) Equation Transcription: Text Transcription: sum_n =1^infinity of ln ( n^2 + 1/ 2n^2 + 1)

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \(\sum_{k=0}^{\infty}(\sqrt{2})^{-k}\) Equation Transcription: Text Transcription: sum _k=0^infinity (sqrt)^-k

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \(\sum_{n=1}^{\infty} \arctan n\) Equation Transcription: Text Transcription: the sum from n =1 to infinity of arctan n

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \(\sum_{n=1}^{\infty}\left(\frac{3}{5^{n}}+\frac{2}{n}\right)\) Equation Transcription: Text Transcription: sum _n=1^infinity (?^n + 2/n)

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \(\sum_{n=1}^{\infty} \frac{2^{n}+4^{n}}{e^{n}}\) Equation Transcription: Text Transcription: sum_n=1^infinity 1/e^n+1/n(n+1)

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \(\sum_{n=1}^{\infty} \frac{e^{n}}{n^{2}}\) Equation Transcription: Text Transcription: sum _n=1^infinity e^n/n^2

Let \(x=0.99999 \ldots\) (a) Do you think that \(x<1\) or \(x=1\)? (b) Sum a geometric series to find the value of \(x\). (c) How many decimal representations does the number 1 have? (d) Which numbers have more than one decimal representation? Equation Transcription: Text Transcription: x=0.99999… x<1 x=1 x

A sequence of terms is defined by \(a_{1}=1 \quad a_{n}=(5-n) a_{n-1}\) Calculate \(\sum_{n-1}^{\infty} a_{n}\). Equation Transcription: Text Transcription: a_1=1 a_n=(5-n)a_n-1 um_n=1^infinity a_n

Express the number as a ratio of integers. \(0 . \overline{8}=0.8888 \ldots\)

Express the number as a ratio of integers. \(0 . \overline{46}=0.46464646 \ldots\) Equation Transcription: ? ? ? Text Transcription: 0.46^_ 0.46464646 . . .

Express the number as a ratio of integers. \(2 . \overline{516}=2.516516516 \ldots\) Equation Transcription: Text Transcription: 2. overline 516=2.516516516 . . .

Express the number as a ratio of integers. \(10.1 \overline{35}=10.135353535 \ldots\) Equation Transcription: Text Transcription: 10.overline 135=10.135353535 . . .

Express the number as a ratio of integers. \(1.234 \overline{567}\) Equation Transcription: Text Transcription: 1.234 overline 567

Express the number as a ratio of integers. \(5 . \overline{71358}\) Equation Transcription: Text Transcription: 5. overline 71358

Find the values of \(x\) for which the series converges. Find the sum of the series for those values of \(x\). \(\sum_{n=1}^{\infty}(-5)^{n} x^{n}\) Equation Transcription: Text Transcription: x sum_n=1^infinity (-5)^n x^n

Find the values of \(x\) for which the series converges. Find the sum of the series for those values of \(x\). \(\sum_{n=1}^{\infty}(x+2)^{n}\) Equation Transcription: Text Transcription: x sum _n=1^infinity (x+2)^n

Find the values of \(x\) for which the series converges. Find the sum of the series for those values of \(x\). \(\sum_{n=0}^{\infty} \frac{(x-2)^{n}}{3^{n}}\) Equation Transcription: Text Transcription: x sum_n=0^infinity (x-2)^n/3^n

Find the values of \(x\) for which the series converges. Find the sum of the series for those values of \(x\). \(\sum_{n=0}^{\infty}(-4)^{n}(x-5)^{n}\) Equation Transcription: Text Transcription: sum_n=0^infinity (-4)^n(x-5)^n

Find the values of \(x\) for which the series converges. Find the sum of the series for those values of \(x\). \(\sum_{n=0}^{\infty} \frac{2^{n}}{x^{n}}\) Equation Transcription: Text Transcription: x sum _n=0^infinity 2^n/x^n

Find the values of \(x\) for which the series converges. Find the sum of the series for those values of \(x\). \(\sum_{n=0}^{\infty} \frac{x^{n}}{2^{n}}\) Equation Transcription: Text Transcription: x sum _n=0^infinity x^n/2^n

Find the values of for which the series converges. Find the sum of the series for those values of .

Find the values of \(x\) for which the series converges. Find the sum of the series for those values of \(x\). \(\sum_{n=0}^{\infty} \frac{\sin ^{n} x}{3^{n}}\) Equation Transcription: Text Transcription: x the sum from n = 0 to infinity of sin^n x/ 3^n x

Use the partial fraction command on a computer algebra system to find a convenient expression for the partial sum, and then use this expression to find the sum of the series. Check your answer by using the CAS to sum the series directly. \(\sum_{n=1}^{\infty} \frac{3 n^{2}+3 n+1}{\left(n^{2}+n\right)^{3}}\) Equation Transcription: Text Transcription: sum _n=1^infinity 3n^2/(n^2 + n)^2

Use the partial fraction command on a computer algebra system to find a convenient expression for the partial sum, and then use this expression to find the sum of the series. Check your answer by using the CAS to sum the series directly.

If the nth partial sum of a series \(\sum_{n=1}^{\infty} a_{n}\) is \(s_{n}=\frac{n-1}{n+1}\) find \(a_{n}\) and \(\sum_{n=1}^{\infty} a_{n}\).

If the \(n\)th partial sum of a series \(\sum_{n=1}^{\infty} a_{n}\) is \(s_{n}=3-n 2^{-n}\), find \(a_{n}\) and \(\sum_{n=1}^{\infty} a_{n}\). Equation Transcription: Text Transcription: n sum _n=1^infinity a sub n s sub n = 3-n2^-n a sub n sum _n=1^infinity a sub n

A doctor prescribes a antibiotic tablet to be taken every eight hours. It is known that the body eliminates of the drug in eight hours. (a) How much of the drug is in the body just after the second tablet is taken? After the third tablet? (b) If is the quantity of the antibiotic in the body just after the th tablet is taken, find an equation that expresses in terms of . (c) What quantity of the antibiotic remains in the body in the long run?

A patient is injected with a drug every 12 hours. Immediately before each injection the concentration of the drug has been reduced by and the new dose increases the concentration by . (a) What is the concentration after three doses? (b) If is the concentration after the th dose, find a formula for as a function of . (c) What is the limiting value of the concentration?

A patient takes 150mg of a drug at the same time every day. It is known that the body eliminates 95% of the drug in 24 hours. (a) What quantity of the drug is in the body after the third tablet? After the nth tablet? (b) What quantity of the drug remains in the body in the long run?

After injection of a dose of insulin, the concentration of insulin in a patient's system decays exponentially and so it can be written as , where represents time in hours and is a positive constant. (a) If a dose is injected every hours, write an expression for the sum of the residual concentrations just before the st injection. (b) Determine the limiting pre-injection concentration. (c) If the concentration of insulin must always remain at or above a critical value , determine a minimal dosage in terms of , and .

When money is spent on goods and services, those who receive the money also spend some of it. The people receiving some of the twice-spent money will spend some of that, and so on. Economists call this chain reaction the multiplier effect. In a hypothetical isolated community, the local government begins the process by spending dollars. Suppose that each recipient of spent money spends and saves of the money that he or she receives. The values and are called the marginal propensity to consume and the marginal propensity to save and, of course, . (a) Let be the total spending that has been generated after transactions. Find an equation for . (b) Show that , where . The number is called the multiplier. What is the multiplier if the marginal propensity to consume is ? Note: The federal government uses this principle to justify deficit spending. Banks use this principle to justify lending a large percentage of the money that they receive in deposits.

A certain ball has the property that each time it falls from a height h onto a hard, level surface, it rebounds to a height rh, where 0 < r < 1. Suppose that the ball is dropped from an initial height of H meters. (a) Assuming that the ball continues to bounce indefinitely, find the total distance that it travels. (b) Calculate the total time that the ball travels. (Use the fact that the ball falls \(\frac{1}{2} g t^{2}\) meters in t seconds.) (c) Suppose that each time the ball strikes the surface with velocity v it rebounds with velocity \(-k v\), where 0 < k < 1. How long will it take for the ball to come to rest?

Find the value of if .

Find the value of such that .

The Harmonic Series Diverges In Example 8 we proved that the harmonic series diverges. Here we outline additional methods of proving this fact. In each case, assume that the series converges with sum , and show that this assumption leads to a contradiction.

The Harmonic Series Diverges In Example 8 we proved that the harmonic series diverges. Here we outline additional methods of proving this fact. In each case, assume that the series converges with sum S, and show that this assumption leads to a contradiction. \(=1+\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)+\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}\right)+\) \(\left(\frac{1}{8}+\frac{1}{9}+\frac{1}{10}\right)+\cdots>1+\frac{3}{3}+\frac{3}{6}+\frac{3}{9}+\cdots=1+S\) Hint: First show that \(\frac{1}{n-1}+\frac{1}{n+1}>\frac{2}{n}\)

The Harmonic Series Diverges In Example 8 we proved that the harmonic series diverges. Here we outline additional methods of proving this fact. In each case, assume that the series converges with sum , and show that this assumption leads to a contradiction. Hint: First show that .

Graph the curves , for , on a common screen. By finding the areas between successive curves, give a geometric demonstration of the fact, shown in Example 2, that

The figure shows two circles and of radius 1 that touch at . The line is a common tangent line; is the circle that touches , and is the circle that touches , and is the circle that touches , and . This procedure can be continued indefinitely and produces an infinite sequence of circles . Find an expression for the diameter of and thus provide another geometric demonstration of Example 2 .

A right triangle is given with and . is drawn perpendicular to is drawn perpendicular to , and this process is continued indefinitely, as shown in the figure. Find the total length of all the perpendiculars in terms of and .

What is wrong with the following calculation? (Guido Ubaldus thought that this proved the existence of God because "something has been created out of nothing.")

Suppose that is known to be a convergent series. Prove that is a divergent series.

(a) Prove part (i) of Theorem 8 . (b) Prove part (iii) of Theorem 8 .

If is divergent and , show that is divergent.

If is convergent and is divergent, show that the series is divergent. [Hint: Argue by contradiction.

If and are both divergent, is necessarily divergent?

Suppose that a series has positive terms and its partial sums satisfy the inequality for all . Explain why must be convergent.

The Fibonacci sequence was defined in Section by the equations Show that each of the following statements is true. (a) (b) (c)

The Cantor set, named after the German mathematician Georg Cantor (1845-1918), is constructed as follows. We start with the closed interval and remove the open interval. That leaves the two intervals and and we remove the open middle third of each. Four intervals remain and again we remove the open middle third of each of them. We continue this procedure indefinitely, at each step removing the open middle third of every interval that remains from the preceding step. The Cantor set consists of the numbers that remain in after all those intervals have been removed. (a) Show that the total length of all the intervals that are removed is 1 . Despite that, the Cantor set contains infinitely many numbers. Give examples of some numbers in the Cantor set. (b) The Sierpinski carpet is a two-dimensional counterpart of the Cantor set. It is constructed by removing the center one-ninth of a square of side 1 , then removing the centers of the eight smaller remaining squares, and so on. (The figure shows the first three steps of the construction.) Show that the sum of the areas of the removed squares is 1 . This implies that the Sierpinski carpet has area

(a) A sequence \(\left\{a_{n}\right\}\) is defined recursively by the equation \(a_{n}=\frac{1}{2}\left(a_{n-1}+a_{n-2}\right)\) for \(n \geq 3\), where \(a_{1}\) and \(a_{2}\) can be any real numbers. Experiment with various values of \(a_{1}\) and \(a_{2}\) and use a calculator to guess the limit of the sequence. (b) Find \(\lim _{n \rightarrow \infty} a_{n}\) in terms of \(a_{1}\) and \(a_{2}\) by expressing \(a_{n+1}-a_{n}\) in terms of \(a_{2}-a_{1}\) and summing a series.

Consider the series \(\sum_{n=1}^{\infty} n /(n+1) !\). (a) Find the partial sums \(s_{1}, s_{2}, s_{3}\) and \(S_{4}\). Do you recognize the denominators? Use the pattern to guess a formula for \(s_{n}\). (b) Use mathematical induction to prove your guess. (c) Show that the given infinite series is convergent, and find its sum. Equation Transcription: Text Transcription: sum _n=1^infinity n/(n + 1)! s sub 1, s sub 2, s sub 3, s sub 4 s sub n

The figure shows infinitely many circles approaching the vertices of an equilateral triangle, each circle touching other circles and sides of the triangle. If the triangle has sides of length 1, find the total area occupied by the circles.