Chapter 12

Show that $$\sum_{k=1}^{\infty}\left(\frac{k+1}{k}\right)^{k} \text { diverges. }$$

(a) Expand \(\sin x\) and \(\cos x\) in powers of \(x-a.\) (b) Show that both series are absolutely convergent for all real \(x.\) (c) As noted earlier (Section 12.5 ), Riemann proved that the order of the terns of an absolutely convergent series that be changed without altering the sum of the series. Use Riemann's discovery and the Taylor expansions of part (a) to derive the addition formulas $$\begin{aligned}&\sin \left(x_{1}+x_{2}\right)=\sin x_{1} \cos x_{2}+\cos x_{1} \sin x_{2}.\\\ &\cos \left(x_{1}+x_{2}\right)=\cos x_{1} \cos x_{2}-\sin x_{1} \sin x_{2}.\end{aligned}$$

Find the Lagrange form of the remainder \(R_{n}(x)\). $$f(x)=\ln (1+x) ; \quad n=5$$

Determine whether the series converges or diverges. $$\sum \frac{k^{k}}{\left(3^{k}\right)^{2}}$$

Estimate to within 0.01 by using series. $$\int_{0}^{1} x^{4} e^{-x^{2}} d x$$

Find the interval of convergence. $$\sum \frac{1}{(\ln k)^{k}}(x-1)^{k}$$

The partial sum indicated is used to estimate the sum of the series. Estimate the error. \(\sum_{k=1}^{\infty}(-1)^{k} \frac{1}{(10)^{k}} ; \quad s_{4}\).

(a) Assume that \(d_{k} \rightarrow 0\) and show that \(\sum_{k=1}^{x}\left(d_{k}-d_{k+1}\right)=e d_{1}\) (b) Sum the following series: (i) \(\sum_{i=1}^{\infty} \frac{\sqrt{k+1}-\sqrt{k}}{\sqrt{k(k+1)}}\) (ii) \(\sum_{k=1}^{\infty} \frac{2^{L}+1}{2 k^{2}(k+1)^{2}}\)

Use a CAS to determine the Taylor polynomial \(P_{6}\) in powers of \((x-1)\) for \(f(x)=\arctan x.\)

Find the interval of convergence. $$\sum(-1)^{k}\left(\frac{2}{3}\right)^{k}(x+1)^{k}$$

Find the Lagrange form of the remainder \(R_{n}(x)\). $$f(x)=\cos 2 x ; \quad n=4$$

Determine whether the series converges or diverges. $$\sum \frac{k^{k}}{3^{k^{2}}}$$

Determine whether the series converges or diverse. $$\sum \frac{2 k}{(2 k) !}$$

Estimate to within 0.01 by using series. $$\int_{0}^{1} \arctan x^{2} d x$$

Show that $$ \sum_{k=1}^{\infty} k x^{k-1}=\frac{1}{(1-x)^{2}} \quad \text { for } \quad|x|<1 $$ HINT: Verify that \(s_{n}\), the \(n\)th partial sum of the series, satisfus the identity $$ (1-x)^{2} s_{n}=1-(n+1) x^{\infty}+n x^{a+1} $$

Use a CAS to determine the Taylor polynomial \(P_{8}\) in powers of \((x-2)\) for \(f(x)=\cosh 2 x.\)

Find the interval of convergence. $$\sum \frac{2^{1 / i} \pi^{k}}{k(k+1)(k+2)}(x-2)^{k}$$

Let \(s_{y}\) be the \(n\) the partial sum of the series \(\sum_{k=0}^{\infty}(-1)^{k} \frac{1}{10 !}.\) Find the least value of \(n\) for which \(s\), approximates the sum of the series within (a) \(0.001 .\) (b) 0.0001.

Determine whether the series converges or diverse. $$\sum \frac{2 k}{(2 k) !}$$

Estimate to within 0.01 by using series. $$\int_{1}^{2} \frac{1-\cos x}{x} d x$$

Speed of convergence) Find the least integer N for which the \(n\)th partial sum of the series differs from the sum of the series by less than 0.0001. $$\sum_{k=1}^{\infty} \frac{1}{4^{i}}$$

Determine whether the series converges or diverges. $$\frac{1}{2}+\frac{2}{3^{2}}+\frac{4}{4^{3}}+\frac{8}{5^{4}}+\cdots$$

Find the interval of convergence. $$1-\frac{x}{2}+\frac{2 x^{2}}{4}-\frac{3 x^{2}}{8}+\frac{4 x^{4}}{16}-\cdots$$

Find the Lagrange form of the remainder \(R_{n}(x)\). $$f(x)=\tan x ; \quad n=2$$

Speed of convergence) Find the least integer N for which the \(n\)th partial sum of the series differs from the sum of the series by less than 0.0001. $$\sum_{k=0}^{\infty}(0.9)^{k}$$

Use a CAS to find the least integer \(n\) for which \(s_{n}\) approximates the sum of the series to the indicated accuracy. Find \(s_{n}.\) \(\sum_{i=1}^{\infty}(-1)^{k} \frac{(0.9)^{i}}{k} ; \quad 0.001.\)

Determine whether the series converges or diverges. $$1+\frac{1 \cdot 2}{1 \cdot 3}+\frac{1 \cdot 2 \cdot 3}{1 \cdot 3 \cdot 5}+\frac{1 \cdot 2 \cdot 3 \cdot 4}{1 \cdot 3 \cdot 5 \cdot 7}+\dots$$

Use a power series to estimate the integral within 0.0001. $$\int_{0}^{0.5} \frac{1-\cos x}{x^{2}} d x$$

Find the interval of convergence. $$\frac{(x-1)}{5^{2}}+\frac{4}{5^{4}}(x-1)^{2}+\frac{9}{5^{6}}(x-1)^{3}+\frac{16}{5^{8}}(x-1)^{4}+\cdots$$

Find the Lagrange form of the remainder \(R_{n}(x)\). $$f(x)=\sin x ; \quad n=5$$

Find the Lagrange form of the remainder \(R_{n}(x)\). $$f(x)=\arctan x ; \quad n=2$$

Determine whether the series converges or diverges. $$\frac{1}{4}+\frac{1 \cdot 3}{4 \cdot 7}+\frac{1 \cdot 3 \cdot 5}{4 \cdot 7 \cdot 10}+\frac{1 \cdot 3 \cdot 5 \cdot 7}{4 \cdot 7 \cdot 10 \cdot 13}+\cdots$$

(a) Show that \(\sum_{h=0}^{\infty} e^{-\alpha A}\) converges for cach \(\alpha > 0\) (b) Show that \(\sum_{k=0}^{\infty} k e^{-\omega k}\) converges for \(\operatorname{cach} \alpha\) ? 0 . (c) Show that, more generally, \(\sum_{k=0}^{\infty} k^{n} e^{-\alpha k}\) converges for cach nonnegative integer \(n\) and \(\operatorname{cach} \alpha > 0\)

Use a power series to estimate the integral within 0.0001. $$\int_{0}^{0.5} \frac{\ln (1+x)}{x} d x$$

Find the interval of convergence. $$\frac{3 x^{2}}{4}+\frac{9 x^{4}}{9}+\frac{27 x^{6}}{16}+\frac{81 x^{8}}{25}+\cdots$$

Speed of convergence) Find the least integer N for which the \(n\)th partial sum of the series differs from the sum of the series by less than 0.0001. $$\sum_{k=0}^{\infty}\left(\frac{2}{3}\right)^{k}$$

Find the Lagrange form of the remainder \(R_{n}(x)\). $$f(x)=\frac{1}{1+x} ; \quad n=4$$

Determine whether the series converges or diverges. $$\frac{2}{3}+\frac{2 \cdot 4}{3 \cdot 7}+\frac{2 \cdot 4 \cdot 6}{3 \cdot 7 \cdot 11}+\frac{2 \cdot 4 \cdot 6 \cdot 8}{3 \cdot 7 \cdot 11 \cdot 15}+\cdots$$

Let \(p > 1 .\) Use the integral test to show that $$\frac{1}{(p-1)(n+1)^{-1}}<\sum_{i=1}^{\infty} \frac{1}{k^{p}}-\sum_{i=1}^{n} \frac{1}{k^{p}} < \frac{1}{(p \sim 1) n^{p-1}}$$

Use a power series to estimate the integral within 0.0001. $$\int_{0}^{02} x \sin x d x$$

Verify that the series \(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{2}-\frac{1}{3}-\frac{1}{4}-\frac{1}{3}-\frac{1}{4}+\frac{1}{3}-\frac{1}{4}+\cdots\) diverges and explain how this does not violate the basic theorem on alternating series.

Start with the geometric series \(\sum_{k=0}^{\infty} x^{i}\) with \(|x|<1\) and a positive number \(\epsilon\). Determine the least positive integer \(N\) for which \(\left|L-s_{N}\right|<\epsilon\) given that the sum of the series is \(L\) and \(s_{N}\) is the Nth partial sum.

Sum the series. $$\sum_{k=9}^{\infty} \frac{1}{k !} x^{3 k}$$

Suppose that the series \(\sum_{k=a}^{\infty} a_{i}(x-1)^{k}\)converges at \(x=3\) What can you conclude about the convergence or divergence of the following series? (a) \(\sum_{k=0}^{\infty} a_{k}\) (b) \(\sum_{n=0}^{\infty}(-1)^{k} a_{k}\) (c) \(\sum_{k=0}^{\infty}(-1)^{k} a_{k} 2^{k}\)

Let \(L\) be the sum of the series \(\sum_{k=9}^{\infty}(-1)^{k} \frac{1}{k !}\) and let \(s_{n}\) be the \(n\) the partial sum. Find the least value of \(n\) for which \(s_{s}\) approximates \(L\) to within (a) \(0.01,\) (b) 0.001.

Sum the series. $$\sum_{k=0}^{\infty} \frac{1}{k !} x^{3 k+1}$$

Suppose that the series \(\sum_{k=0}^{\infty} a_{k}(x+2)^{k}\) converges at \(x=4\) At what other values of \(x\) must the series converge? Does the series necessarily converge at \(x=-8 ?\)

Let \(a_{0}, a_{1}, a_{2}, \cdots\) be a non-increasing sequence of positive numbers that converges to \(0 .\) Does the alternating series \(\sum(-1)^{k} a_{k}\) necessity converge?

Complete the proof of the ratio test by proving that (a) if \(\lambda>1,\) then \(\sum a_{k}\) diverges, and (b) if \(\lambda=1,\) the ratio test is inconclusive.

(a) Use a CAS or graphing utility. Calculate the sum of the first 100 terns of the series. (b) Use the inequalities given in Exercise 40 to obtain upper and lower bounds for \(R_{100} .\) (c) Use parts (a) and (b) to estimate the sum of the series. $$\sum_{k=1}^{\infty} \frac{1}{k^{4}}$$