Chapter 30

For each series in Problems 1 through 9 , determine whether the series converges absolutely, converges conditionally, or diverges. $$ \sum_{k=1}^{\infty}(-1)^{k} \frac{k !}{(k-1) !} $$

Find a good upper bound for the magnitude of the error involved in approximating \(\cos x\) by \(1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}\) for \(|x| \leq 0.2\). Do this using Taylor s Inequality; then check your answer by graphing the remainder function.

A power series centered at \(b=0\) has a radius of convergence of \(5 .\) For each value of \(x\) given below, determine whether the series converges, diverges, or there is not enough information available to determine. (a) \(x=0\) (b) \(x=3\) (c) \(x=5\) (d) \(x=7\) (e) \(x=-1.8\) (f) \(x=-\sqrt{5}\) (g) \(x=-5\) (h) \(x=-6\)

Find the Maclaurin series for \(\cos x\) and show that it is equal to \(\cos x\) for all \(x\).

Do the following. (a) Compute the fourth degree Taylor polynomial for \(f(x)\) at \(x=0 .\) (b) On the same set of axes, graph \(f(x), P_{1}(x), P_{2}(x), P_{3}(x)\), and \(P_{4}(x)\). (c) Use \(P_{1}(x), P_{2}(x), P_{3}(x)\), and \(P_{4}(x)\) to approximate \(f(0.1)\) and \(f(0.3) .\) Compare these approximations to those given by a calculator. $$ f(x)=e^{-x} $$

For each series, determine whether the series converges absolutely, converges conditionally, or diverges. $$ \sum_{k=1}^{\infty}(-1)^{k+1} \frac{k !}{(k+1) !} $$

Use the third degree Taylor polynomial for \(e^{x}\) at \(x=0\) to estimate \(\sqrt{e}\). Then use Taylor \(s\) Theorem to get a reasonable upper bound for the remainder.

(a) Find the Maclaurin series for \(\ln (1+x)\). (b) On the same set of axes, graph \(\ln (1+x)\) and \(P_{6}(x)\). Observe that the polynomial approximation to \(\ln (1+x)\) is good for \(|x|<1\). (c) Graph \(R_{6}(x)=\ln (1+x)-P_{6}(x)\). Observe that \(R_{6}(x)\) is close to zero on \(|x|<1\). In the next section we will show that the radius of convergence of the Maclaurin series for \(\ln (1+x)\) is 1 .

Do the following. (a) Compute the fourth degree Taylor polynomial for \(f(x)\) at \(x=0 .\) (b) On the same set of axes, graph \(f(x), P_{1}(x), P_{2}(x), P_{3}(x)\), and \(P_{4}(x)\). (c) Use \(P_{1}(x), P_{2}(x), P_{3}(x)\), and \(P_{4}(x)\) to approximate \(f(0.1)\) and \(f(0.3) .\) Compare these approximations to those given by a calculator. $$ f(x)=\ln (1+x) $$

Suppose \(0 \leq a_{k} \leq b_{k} \leq c_{k}\) for all \(k .\) Consider \(\sum_{k=1}^{\infty} a_{k}, \sum_{k=1}^{\infty} b_{k}\) and \(\sum_{k=1}^{\infty} c_{k} .\) What conclusions can be drawn if you know that \(\sum_{k=1}^{\infty} b_{k}\) (a) converges. (b) diverges.

For each series, determine whether the series converges absolutely, converges conditionally, or diverges. $$ \sum_{k=1}^{\infty}(-1)^{k} \frac{1}{3 k} $$

Do the following. (a) Compute the fourth degree Taylor polynomial for \(f(x)\) at \(x=0 .\) (b) On the same set of axes, graph \(f(x), P_{1}(x), P_{2}(x), P_{3}(x)\), and \(P_{4}(x)\). (c) Use \(P_{1}(x), P_{2}(x), P_{3}(x)\), and \(P_{4}(x)\) to approximate \(f(0.1)\) and \(f(0.3) .\) Compare these approximations to those given by a calculator. $$ f(x)=\tan ^{-1} x $$

In Problems 4 through 19, determine whether the series converges or diverges. It is possible to solve Problems 4 through 19 without the Limit Comparison, Ratio, and Root Tests. \(\sum_{k=2}^{\infty} \frac{3}{\sqrt{k}}\)

For each series, determine whether the series converges absolutely, converges conditionally, or diverges. $$ \sum_{k=2}^{\infty}(-1)^{n} \frac{k}{\ln k} $$

Use the third degree Taylor polynomial for \(\ln x\) centered at \(x=1,(x-1)-\frac{(x-1)^{2}}{2}+\) \(\frac{(x-1)^{3}}{3}\), to approximate \(\ln (1.5) .\) Then give an upper bound for the remainder using Taylor s Theorem.

Find the Taylor series for \(f(x)\) centered at the indicated value of \(b\). $$ f(x)=\sin x, \quad b=\pi $$

Do the following. (a) Compute the fourth degree Taylor polynomial for \(f(x)\) at \(x=0 .\) (b) On the same set of axes, graph \(f(x), P_{1}(x), P_{2}(x), P_{3}(x)\), and \(P_{4}(x)\). (c) Use \(P_{1}(x), P_{2}(x), P_{3}(x)\), and \(P_{4}(x)\) to approximate \(f(0.1)\) and \(f(0.3) .\) Compare these approximations to those given by a calculator. $$ f(x)=(1+x)^{4} $$

Determine whether the series converges or diverges. It is possible to solve Problems 4 through 19 without the Limit Comparison, Ratio, and Root Tests. \(\sum_{k=10}^{\infty} \frac{10}{k \sqrt{k}}\)

For each series, determine whether the series converges absolutely, converges conditionally, or diverges. $$ \sum_{k=10}^{\infty} \frac{\cos (k n)}{10 k} $$

The second degree Taylor polynomial for \(f(x)=(1+x)^{p}\) is \(1+p x+\frac{p(p-1)}{2 !} x^{2} .\) If the second degree Taylor polynomial is used to approximate \(\sqrt{1+x}\) for \(|x| \leq 0.2\), nd an upper bound for the magnitude of the error. Use the Taylor Inequality; then check your answer by graphing \(R_{2}(x)\).

Find the Taylor series for \(f(x)\) centered at the indicated value of \(b\). $$ f(x)=2 \cos x, \quad b=\frac{\pi}{2} $$

Do the following. (a) Compute the fourth degree Taylor polynomial for \(f(x)\) at \(x=0 .\) (b) On the same set of axes, graph \(f(x), P_{1}(x), P_{2}(x), P_{3}(x)\), and \(P_{4}(x)\). (c) Use \(P_{1}(x), P_{2}(x), P_{3}(x)\), and \(P_{4}(x)\) to approximate \(f(0.1)\) and \(f(0.3) .\) Compare these approximations to those given by a calculator. $$ f(x)=\sqrt{1+x} $$

Determine whether the series converges or diverges. It is possible to solve Problems 4 through 19 without the Limit Comparison, Ratio, and Root Tests. \(\sum_{k=1}^{\infty} \frac{\ln k}{k}\)

For each series, determine whether the series converges absolutely, converges conditionally, or diverges. $$ \sum_{k=0}^{\infty}\left(-\frac{11}{12}\right)^{k} $$

For \(x\) near zero, \(\cos x \approx 1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}-\cdots+(-1)^{n} \frac{x^{2 n}}{(2 n) !} .\) What degree Taylor polyno- mial must be used to approximate \(\cos (0.2)\) with error less than \(\frac{1}{10^{8}}\) ?

Find the Taylor series for \(f(x)\) centered at the indicated value of \(b\). $$ f(x)=10^{x}, \quad b=0 $$

Do the following. (a) Compute the fourth degree Taylor polynomial for \(f(x)\) at \(x=0 .\) (b) On the same set of axes, graph \(f(x), P_{1}(x), P_{2}(x), P_{3}(x)\), and \(P_{4}(x)\). (c) Use \(P_{1}(x), P_{2}(x), P_{3}(x)\), and \(P_{4}(x)\) to approximate \(f(0.1)\) and \(f(0.3) .\) Compare these approximations to those given by a calculator. $$ f(x)=2 x^{4}-3 x^{2}+x-1 $$

Determine whether the series converges or diverges. It is possible to solve Problems 4 through 19 without the Limit Comparison, Ratio, and Root Tests. \(\sum_{n=5}^{\infty} n^{-9 / 10}\)

For each series, determine whether the series converges absolutely, converges conditionally, or diverges. $$ \sum_{k=1}^{\infty} \frac{1}{100} \sin \left(\frac{k \pi}{2}\right) $$

Approximate \(\sqrt[3]{27.5}\) using an appropriate second degree Taylor polynomial. Find a good upper bound for the error by using Taylor s Inequality.

Find the Taylor series for \(f(x)\) centered at the indicated value of \(b\). $$ f(x)=\frac{1}{\sqrt{x}}, \quad b=1 $$

Do the following. (a) Compute the fourth degree Taylor polynomial for \(f(x)\) at \(x=0 .\) (b) On the same set of axes, graph \(f(x), P_{1}(x), P_{2}(x), P_{3}(x)\), and \(P_{4}(x)\). (c) Use \(P_{1}(x), P_{2}(x), P_{3}(x)\), and \(P_{4}(x)\) to approximate \(f(0.1)\) and \(f(0.3) .\) Compare these approximations to those given by a calculator. $$ f(x)=(1+x)^{-2} $$

Determine whether the series converges or diverges. It is possible to solve Problems 4 through 19 without the Limit Comparison, Ratio, and Root Tests. \(\sum_{k=1}^{\infty} \frac{k}{e^{k}}\)

For each series, determine whether the series converges absolutely, converges conditionally, or diverges. $$ \sum_{k=1}^{\infty}(-1)^{k} \frac{2^{k}}{k} $$

The second degree Taylor polynomial generated by \(\ln (1+x)\) about \(x=0\) is \(x-\frac{x^{2}}{2}\). Use Taylor s Theorem to nd a good upper bound on the error involved in using this polynomial to approximate the following. (a) \(\ln (1.2)\) (b) \(\ln (0.8)\)

Find the Taylor series for \(f(x)\) centered at the indicated value of \(b\). $$ f(x)=(3+2 x)^{3}, \quad b=0 $$

Determine whether the series converges or diverges. It is possible to solve Problems 4 through 19 without the Limit Comparison, Ratio, and Root Tests. \(\sum_{k=2}^{\infty} 2 k^{-10 / 9}\)

For each series, determine whether the series converges absolutely, converges conditionally, or diverges. $$ \sum_{k=0}^{\infty}(-1)^{n}\left(\frac{k^{2}-10}{2 k^{2}+5 k}\right) $$

Find the Taylor series for \(f(x)\) centered at the indicated value of \(b\). $$ f(x)=(1+x)^{5}, \quad b=0 $$

Determine whether the series converges or diverges. It is possible to solve Problems 4 through 19 without the Limit Comparison, Ratio, and Root Tests. \(\sum_{k=1}^{\infty} e^{-2 k}\)

Is it possible for a geometric series to converge conditionally? If it is possible, produce an example.

(a) Find the second degree Taylor polynomial generated by sec \(x\) at \(x=0\). (b) Graph \(P_{2}(x)\) and sec \(x\) on the same set of axes.

Determine whether the series converges or diverges. It is possible to solve Problems 4 through 19 without the Limit Comparison, Ratio, and Root Tests. \(\sum_{k=1}^{\infty} \frac{k+2}{3 k^{2}}\)

A hyena is loping down a straight path away from a stream. The hyena is \(6 \mathrm{~m}\) from the stream, moving at a rate of \(2 \mathrm{~m} / \mathrm{s}\) and decelerating at a rate of \(0.1 \mathrm{~m} / \mathrm{s}^{2}\). Use a second degree Taylor polynomial to estimate its distance from the stream 1 second later.

A power series of the form \(\sum_{k=0}^{\infty} a_{k}(x-2)^{k}\) has a radius of convergence of \(3 .\) (a) For what values of \(x\) can you say with con dence that the series converges? (b) For what values of \(x\) can you say with con dence that the series diverges? (c) For what values of \(x\) are you given inadequate information to determine convergence?

(a) Compute the third degree Taylor polynomial for \(\tan x\) about \(x=0\). (b) Why is it reasonable to expect the coef cient of the \(x^{2}\) term to be zero?

Determine whether the series converges or diverges. It is possible to solve Problems 4 through 19 without the Limit Comparison, Ratio, and Root Tests. \(\sum_{k=1}^{\infty} 2 e^{-0.1 k}\)

Approximate \(\frac{1}{e}\) with error less than \(10^{-5}\).

What degree Taylor polynomial for \(e^{x}\) about \(x=0\) must be used to approximate \(e^{0.3}\) with error less than \(10^{-5}\) ?

The interval of convergence of a power series is \((-2,5]\). (a) What is the radius of convergence? (b) What is the center of the series?