Loglog p-series Consider the series , where p is a real

Explain why computation alone may not determine whether a series converges.

31-34. p-series Determine the convergence or divergence of the following series. \(\sum_{k=2}^{\infty} \frac{k^{e}}{k^{\pi}}\)

Stacking dominoes Consider a set of identical dominoes that are 2 in long. The dominoes are stacked on top of each other with their long edges aligned so that each domino overhangs the one beneath it as far as possible (see figure). a. If there are n dominoes in the stack, what is the greatest distance that the top domino can be made to overhang the bottom domino? (Hint: Put the nth domino beneath the previous n - 1 dominoes.) b. If we allow for infinitely many dominoes in the stack, what is the greatest distance that the top domino can be made to overhang the bottom domino?

15-22. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. \(\sum_{k=0}^{\infty} \frac{k}{2 k+1}\)

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If \(\sum_{k=1}^{\infty} a_{k}\) converges, then \(\sum_{k=10}^{\infty} a_{k}\) converges. b. If \(\sum_{k=1}^{\infty} a_{k}\) diverges, then \(\sum_{k=10}^{\infty} a_{k}\) diverges. c. If \(\sum a_{k}\) converges, then \(\sum\left(a_{k}+0.0001\right)\) also converges. d. If \(\sum p^{k}\) diverges, then \(\sum(p+0.001)^{k}\) diverges, for a fixed real number p. e. If \(\sum k^{-p}\) converges, then \(\sum k^{-p+0.001}\) converges. f. If \(\lim_{k\rightarrow\infty}\ a_k=0\), then \(\sum a_{k}\) converges.

35-42. Remainders and estimates Consider the following convergent series. a. Find an upper bound for the remainder in terms of n. b. Find how many terms are needed to ensure that the remainder is less than \(10^{-3}\). c. Find lower and upper bounds (\(L_{n}\) and \(U_{n}\), respectively) on the exact value of the series. d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series. \(\sum_{k=1}^{\infty} \frac{1}{3^{k}}\)

23-30. Integral Test Use the Integral Test to determine the convergence or divergence of the following series. Check that the conditions of the test are satisfied. \(\sum_{k=1}^{\infty} k e^{-2 k^{2}}\)

23-30. Integral Test Use the Integral Test to determine the convergence or divergence of the following series. Check that the conditions of the test are satisfied. \(\sum_{k=1}^{\infty} \frac{k}{e^{k}}\)

35-42. Remainders and estimates Consider the following convergent series. a. Find an upper bound for the remainder in terms of n. b. Find how many terms are needed to ensure that the remainder is less than \(10^{-3}\). c. Find lower and upper bounds (\(L_{n}\) and \(U_{n}\), respectively) on the exact value of the series. d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series. \(\sum_{k=1}^{\infty} k e^{-k^{2}}\)

Loglog p-series Consider the series \(\sum_{k=2}^{\infty} \frac{1}{k \ln k(\ln \ln k)^{p}}\), where p is a real number. a. For what values of p does this series converge? b. Which of the following series converges faster? Explain. \(\sum_{k=2}^{\infty} \frac{1}{k(\ln k)^{2}}\) or \(\sum_{k=2}^{\infty} \frac{1}{k \ln k(\ln \ln k)^{2}}\)?

Property of divergent series Prove that if \(\sum a_{k}\) diverges, then \(\sum c a_{k}\) also diverges, where \(c \neq 0\) is a constant.

Reciprocals of odd squares Given that \(\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6}\) (Exercises 57 and 58) and that the terms of this series may be rearranged without changing the value of the series. Determine the sum of the reciprocals of the squares of the odd positive integers.

Define the remainder of an infinite series.

9-14. Properties of series Use the properties of infinite series to evaluate the following series. \(\sum_{k=0}^{\infty} \frac{2-3^{k}}{6^{k}}\)

15-22. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. \(\sum_{k=1}^{\infty} \frac{\sqrt{k^{2}+1}}{k}\)

44–49. Choose your test Determine whether the following series converge or diverge. \(\sum_{k=0}^{\infty} \frac{k}{\sqrt{k^{2}+4}}\)

A divergence proof Give an argument, similar to that given in the text for the harmonic series, to show that \(\sum_{k=1}^{\infty} \frac{1}{\sqrt{k}}\) diverges.

23-30. Integral Test Use the Integral Test to determine the convergence or divergence of the following series. Check that the conditions of the test are satisfied. \(\sum_{k=1}^{\infty} \frac{1}{\sqrt[3]{k+10}}\)

15-22. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. \(\sum_{k=2}^{\infty} \frac{\sqrt{k}}{\ln ^{10} k}\)

If a series of positive terms converges, does it follow that the remainder \(R_{n}\) must decrease to zero as \(n\ \rightarrow\ \infty\)? Explain.

23-30. Integral Test Use the Integral Test to determine the convergence or divergence of the following series. Check that the conditions of the test are satisfied. \(\sum_{k=0}^{\infty} \frac{1}{\sqrt{k+8}}\)

Explain why the sequence of partial sums for a series with positive terms is an increasing sequence.

23-30. Integral Test Use the Integral Test to determine the convergence or divergence of the following series. Check that the conditions of the test are satisfied. \(\sum_{k=2}^{\infty} \frac{1}{k \ln k}\)

44–49. Choose your test Determine whether the following series converge or diverge. \(\sum_{k=1}^{\infty} \frac{2^{k}+3^{k}}{4^{k}}\)

Is it true that if the terms of a series of positive terms decrease to zero, then the series converges? Explain using an example.

31-34. p-series Determine the convergence or divergence of the following series. \(\sum_{k=1}^{\infty} \frac{1}{k^{10}}\)

The zeta function The Riemann zeta function is the subject of extensive research and is associated with several renowned unsolved problems. It is defined by \(\zeta(x)=\sum_{k=1}^{\infty} \frac{1}{k^{x}}\). When x is a real number, the zeta function becomes a p-series. For even positive integers p, the value of \(\zeta(p)\) is known exactly. For example, \(\sum_{k=1}^{\infty}\frac{1}{k^2}=\frac{\pi^2}{6},\ \ \ \ \quad\sum_{k=1}^{\infty}\frac{1}{k^4}=\frac{\pi^4}{90},\ \ \ \quad\sum_{k=1}^{\infty}\frac{1}{k^6}=\frac{\pi^6}{945},\ldots\) Use the estimation techniques described in the text to approximate \(\zeta(3)\) and \(\zeta(5)\) (whose values are not known exactly) with a remainder less than \(10^{-3}\).

15-22. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. \(\sum_{k=1}^{\infty} \frac{k^{3}}{k^{3}+1}\)

9-14. Properties of series Use the properties of infinite series to evaluate the following series. \(\sum_{k=1}^{\infty}\left[\frac{1}{3}\left(\frac{5}{6}\right)^{k}+\frac{3}{5}\left(\frac{7}{9}\right)^{k}\right]\)

44–49. Choose your test Determine whether the following series converge or diverge. \(\sum_{k=1}^{\infty} \frac{1}{(3 k+1)(3 k+4)}\)

35-42. Remainders and estimates Consider the following convergent series. a. Find an upper bound for the remainder in terms of n. b. Find how many terms are needed to ensure that the remainder is less than \(10^{-3}\). c. Find lower and upper bounds (\(L_{n}\) and \(U_{n}\), respectively) on the exact value of the series. d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series. \(\sum_{k=2}^{\infty} \frac{1}{k(\ln k)^{2}}\)

15-22. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. \(\sum_{k=1}^{\infty} \frac{k}{k^{2}+1}\)

Prime numbers The prime numbers are those positive integers that are divisible by only 1 and themselves (for example, 2, 3, 5, 7, 11, 13,...). A celebrated theorem states that the sequence of prime numbers \(\left\{p_{k}\right\}\) satisfies \(\lim_{k\rightarrow\infty}\ p_k/(k\ln k)=1\). Show that \(\sum_{k=2}^{\infty} \frac{1}{k \ln k}\) diverges, which implies that the series \(\sum_{k=1}^{\infty} 1 / p_{k}\) diverges.

35-42. Remainders and estimates Consider the following convergent series. a. Find an upper bound for the remainder in terms of n. b. Find how many terms are needed to ensure that the remainder is less than \(10^{-3}\). c. Find lower and upper bounds (\(L_{n}\) and \(U_{n}\), respectively) on the exact value of the series. d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series. \(\sum_{k=1}^{\infty} e^{-k}\)

23-30. Integral Test Use the Integral Test to determine the convergence or divergence of the following series. Check that the conditions of the test are satisfied. \(\sum_{k=2}^{\infty} \frac{1}{k(\ln k)^{2}}\)

44–49. Choose your test Determine whether the following series converge or diverge. \(\sum_{k=0}^{\infty} \frac{10}{k^{2}+9}\)

31-34. p-series Determine the convergence or divergence of the following series. \(\sum_{k=1}^{\infty} 2 k^{-3 / 2}\)

9-14. Properties of series Use the properties of infinite series to evaluate the following series. \(\sum_{k=0}^{\infty}\left[\frac{1}{2}(0.2)^{k}+\frac{3}{2}(0.8)^{k}\right]\)

44–49. Choose your test Determine whether the following series converge or diverge. \(\sum_{k=1}^{\infty} \sqrt{\frac{k+1}{k}}\)

9-14. Properties of series Use the properties of infinite series to evaluate the following series. \(\sum_{k=1}^{\infty}\left[\left(\frac{1}{6}\right)^{k}+\left(\frac{1}{3}\right)^{k-1}\right]\)

15-22. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. \(\sum_{k=0}^{\infty} \frac{1}{1000+k}\)

Showing that \(\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6}\) In 1734, Leonhard Euler informally proved that \(\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6}\). An elegant proof is outlined here that uses the inequality. \(\cot ^{2} x<\frac{1}{x^{2}}<1+\cot ^{2} x\) (provided that \(0<x<\pi / 2\)) and the identity \(\sum_{k=1}^{n} \cot ^{2}(k \theta)=\frac{n(2 n-1)}{3}\), for n = 1, 2, 3 . . ., where \(\theta=\frac{\pi}{2 n+1}\). a. Show that \(\sum_{k=1}^{n} \cot ^{2}(k \theta)<\frac{1}{\theta^{2}} \sum_{k=1}^{n} \frac{1}{k^{2}}<n+\sum_{k=1}^{n} \cot ^{2}(k \theta)\). b. Use the inequality in part (a) to show that \(\frac{n(2 n-1) \pi^{2}}{3(2 n+1)^{2}}<\sum_{k=1}^{n} \frac{1}{k^{2}}<\frac{n(2 n+2) \pi^{2}}{3(2 n+1)^{2}}\). c. Use the Squeeze Theorem to conclude that \(\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6}\). [Source: The College Mathematics Journal, 24, No. 5 (November, 1993).]

A sequence of sums Consider the sequence \(\left\{x_{n}\right\}\) defined for n = 1, 2, 3, . . . by \(x_{n}=\sum_{k=n+1}^{2 n} \frac{1}{k}=\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2 n}\). a. Write out the terms \(x_1,\ x_2,\ x_3\). b. Show that \(\frac{1}{2}\le\ x_n<1\) for n = 1, 2, 3, . . . . c. Show that \(x_{n}\) is the right Riemann sum for \(\int_{1}^{2} \frac{d x}{x}\) using n subintervals. d. Conclude that \(\lim_{n\rightarrow\infty}\ x_n=\ln2\).

For what values of p does the series \(\sum_{k=1}^{\infty} \frac{1}{k^{p}}\) converge? For what values of p does it diverge?

9-14. Properties of series Use the properties of infinite series to evaluate the following series. \(\sum_{k=0}^{\infty}\left[3\left(\frac{2}{5}\right)^{k}-2\left(\frac{5}{7}\right)^{k}\right]\)

Shifted p-series Consider the sequence \(\left\{F_{n}\right\}\) defined by \(F_{n}=\sum_{k=1}^{\infty} \frac{1}{k(k+n)}\), for n = 0, 1, 2, .... When n = 0, the series is a p-series, and we have \(F_{0}=\pi^{2} / 6\) (Exercises 57 and 58). a. Explain why \(\left\{F_{n}\right\}\) is a decreasing sequence. b. Plot approximations to \(\left\{F_{n}\right\}\) for n = 1, 2, . . ., 20. c. Based on your experiments, make a conjecture about \(\lim_{n\rightarrow\infty}\ F_n\).

9-14. Properties of series Use the properties of infinite series to evaluate the following series. \(\sum_{k=1}^{\infty}\left[2\left(\frac{3}{5}\right)^{k}+3\left(\frac{4}{9}\right)^{k}\right]\)

Log p-series Consider the series \(\sum_{k=2}^{\infty} \frac{1}{k(\ln k)^{p}}\), where p is a real number. a. Use the Integral Test to determine the values of p for which this series converges. b. Does this series converge faster for p = 2 or p = 3? Explain.

For what values of p does the series \(\sum_{k=10}^{\infty} \frac{1}{k^{p}}\) converge (initial index is 10)? For what values of p does it diverge?

35-42. Remainders and estimates Consider the following convergent series. a. Find an upper bound for the remainder in terms of n. b. Find how many terms are needed to ensure that the remainder is less than \(10^{-3}\). c. Find lower and upper bounds (\(L_{n}\) and \(U_{n}\), respectively) on the exact value of the series. d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series. \(\sum_{k=1}^{\infty} \frac{1}{k^{6}}\)

15-22. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. \(\sum_{k=1}^{\infty} \frac{k^{2}}{2^{k}}\)

15-22. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. \(\sum_{k=2}^{\infty} \frac{k}{\ln k}\)

44–49. Choose your test Determine whether the following series converge or diverge. \(\sum_{k=2}^{\infty} \frac{4}{k \ln ^{2} k}\)

23-30. Integral Test Use the Integral Test to determine the convergence or divergence of the following series. Check that the conditions of the test are satisfied. \(\sum_{k=2}^{\infty} \frac{1}{k \ln k \ln (\ln k)}\)

35-42. Remainders and estimates Consider the following convergent series. a. Find an upper bound for the remainder in terms of n. b. Find how many terms are needed to ensure that the remainder is less than \(10^{-3}\). c. Find lower and upper bounds (\(L_{n}\) and \(U_{n}\), respectively) on the exact value of the series. d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series. \(\sum_{k=1}^{\infty} \frac{1}{k^{8}}\)

35-42. Remainders and estimates Consider the following convergent series. a. Find an upper bound for the remainder in terms of n. b. Find how many terms are needed to ensure that the remainder is less than \(10^{-3}\). c. Find lower and upper bounds (\(L_{n}\) and \(U_{n}\), respectively) on the exact value of the series. d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series. \(\sum_{k=1}^{\infty} \frac{1}{k^{3}}\)

Can the Integral Test be used to determine whether a series diverges?

31-34. p-series Determine the convergence or divergence of the following series. \(\sum_{k=3}^{\infty} \frac{1}{(k-2)^{4}}\)

Find a series Find a series that… a. converges faster than \(\sum \frac{1}{k^{2}}\) but slower than \(\sum \frac{1}{k^{3}}\). b. diverges faster than \(\sum \frac{1}{k}\) but slower than \(\sum \frac{1}{\sqrt{k}}\). c. converges faster than \(\sum \frac{1}{k \ln ^{2} k}\) but slower than \(\sum \frac{1}{k^{2}}\).

23-30. Integral Test Use the Integral Test to determine the convergence or divergence of the following series. Check that the conditions of the test are satisfied. \(\sum_{k=1}^{\infty} \frac{k}{\sqrt{k^{2}+4}}\)

35-42. Remainders and estimates Consider the following convergent series. a. Find an upper bound for the remainder in terms of n. b. Find how many terms are needed to ensure that the remainder is less than \(10^{-3}\). c. Find lower and upper bounds (\(L_{n}\) and \(U_{n}\), respectively) on the exact value of the series. d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series. \(\sum_{k=1}^{\infty} \frac{1}{k^{3 / 2}}\)

Properties proof Use the ideas in the proof of Property 1 of Theorem 8.8 to prove Property 2 of Theorem 8.8.

The harmonic series and Euler's constant a. Sketch the function f(x) = 1/x on the interval [1, n + 1], where n is a positive integer. Use this graph to verify that \(\ln (n+1)<1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}<1+\ln n\) b. Let \(S_{n}\) be the sum of the first n terms of the harmonic series, so part (a) says \(\ln (n+1)<S_{n}<1+\ln n\). Define the new sequence \(\left\{E_{n}\right\}\), where \(E_n=S_n-\ln(n+1)\), for n = 1, 2, 3, . . . . Show that \(E_{n}>0\) for n = 1, 2, 3, . . . . c. Using a figure similar to that used in part (a), show that \(\frac{1}{n+1}>\ln (n+2)-\ln (n+1)\) d. Use parts (a) and (c) to show that \(\left\{E_{n}\right\}\) is an increasing sequence \(\left(E_{n+1}>E_{n}\right)\). e. Use part (a) to show that \(\left\{E_{n}\right\}\) is bounded above by 1. f. Conclude from parts (d) and (e) that \(\left\{E_{n}\right\}\) has a limit less than or equal to 1. This limit is known as Euler's constant and is denoted \(\gamma\) (the Greek lowercase letter gamma). g. By computing terms of \(\left\{E_{n}\right\}\), estimate the value of \(\gamma\) and compare it to the value \(\gamma \approx 0.5772\). (It has been conjectured, but not proved, that \(\gamma\) is irrational.) h. The preceding arguments show that the sum of the first n terms of the harmonic series satisfy \(S_{n} \approx 0.5772+\ln (n+1)\). How many terms must be summed for the sum to exceed 10?