Chapter 30

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

Arrive at the series for \(\cos x\) by differentiating the Maclaurin series for \(\sin x\).

(a) Find the \(n\) th degree Taylor polynomial for \(f(x)=\frac{1}{1-x}\) centered at \(x=0\). (b) How many nonzero terms of the polynomial in part (a) must be used to approximate \(f\left(\frac{1}{2}\right)\) with error less than \(10^{-5}\) ?

A power series is of the form \(\sum_{k=0}^{\infty} a_{k}(x+3)^{k} .\) Which of the intervals given below could conceivably be the interval of convergence of the series? For each option ruled out, explain the rationale. (a) \((0, \infty)\) (b) \((2,4)\) (c) \([-10,4)\) (d) \([-3,3]\) (e) \((-4,2)\) (f) \((-5,-1]\) (g) \((-\infty, \infty)\)

Let \(f(x)=\ln (1+x)\). Find the \(n\) th degree Taylor polynomial generated by \(f\) about \(x=0 .\)

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

Find the Maclaurin series for \(\arcsin x\) using the fact that \(\int \frac{1}{\sqrt{1-x^{2}}} d x=\sin ^{-1} x+C\). What is the radius of convergence of the series?

According to Einstein s theory of special relativity, the mass of an object moving with velocity \(v \mathrm{~m} / \mathrm{s}\) is given by $$ m=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}} $$ where \(m_{0}\) is the mass of the object at rest and \(c\) is the speed of light, \(c=3 \times 10^{8} \mathrm{~m} / \mathrm{s}\). (a) Use the rst degree Taylor polynomial for \(\frac{1}{\sqrt{1+x}}\) to arrive at the estimate $$ m \approx m_{0}+\frac{m_{0}}{2} \frac{v^{2}}{c^{2}} $$ (b) If an object is moving at \(100 \mathrm{~m} / \mathrm{s}\), nd an upper bound for the error involved in using the approximation given in part (a).

Use your knowledge of the binomial series to find the \(n\) th degree Taylor polynomial for \(f(x)\) about \(x=0 .\) Give the radius of convergence of the corresponding Maclaurin series. One of these "series" converges for all \(x\). $$ f(x)=\sqrt{1+3 x}, \quad n=3 $$

Compute the \(n\) th degree Taylor polynomial expansion of \(f(x)=\frac{1}{x}\) about \(x=1\). Graph \(f\) and \(P_{1}, P_{2}, P_{3}\), and \(P_{4}\) on a common 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. \(\sum_{k=1}^{\infty} \frac{k}{k^{3}+k+1}\)

Use your knowledge of the binomial series to find the \(n\) th degree Taylor polynomial for \(f(x)\) about \(x=0 .\) Give the radius of convergence of the corresponding Maclaurin series. One of these "series" converges for all \(x\). $$ f(x)=\frac{1}{\sqrt{1+x}}, \quad n=2 $$

Use a second degree Taylor polynomial centered appropriately to approximate the expression given. $$ \sqrt[3]{8.3} $$

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

Use your knowledge of the binomial series to find the \(n\) th degree Taylor polynomial for \(f(x)\) about \(x=0 .\) Give the radius of convergence of the corresponding Maclaurin series. One of these "series" converges for all \(x\). $$ f(x)=(1-x)^{\frac{2}{3}}, \quad n=3 $$

Use a second degree Taylor polynomial centered appropriately to approximate the expression given. $$ \sqrt{103} $$

Determine whether the series converges or diverges. It is possible to solve Problems 4 through 19 without the Limit Comparison, Ratio, and Root Tests. \(\sum_{k=2}^{\infty} \frac{5}{k-0.5}\)

Write the given integral as a power series. \(\int \frac{1}{1+x^{5}} d x\)

Use your knowledge of the binomial series to find the \(n\) th degree Taylor polynomial for \(f(x)\) about \(x=0 .\) Give the radius of convergence of the corresponding Maclaurin series. One of these "series" converges for all \(x\). $$ f(x)=\sqrt[3]{1+x^{2}}, \quad n=5 $$

Use a second degree Taylor polynomial centered appropriately to approximate the expression given. $$ \tan ^{-1}(0.75) $$

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

Approximate \(\int_{0}^{0.5} \sin \left(x^{2}\right) d x\) with error less than \(10^{-8} .\) Is your approximation an overestimate, or an underestimate?

Use your knowledge of the binomial series to find the \(n\) th degree Taylor polynomial for \(f(x)\) about \(x=0 .\) Give the radius of convergence of the corresponding Maclaurin series. One of these "series" converges for all \(x\). $$ f(x)=(1+3 x)^{5}, \quad n=6 $$

Use a second degree Taylor polynomial centered appropriately to approximate the expression given. $$ \sqrt[3]{29} $$

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

Approximate \(\int_{0}^{0.1} \frac{x}{1+x^{3}} d x\) with error less than \(10^{-10}\).

Use your knowledge of the binomial series to find the \(n\) th degree Taylor polynomial for \(f(x)\) about \(x=0 .\) Give the radius of convergence of the corresponding Maclaurin series. One of these "series" converges for all \(x\). $$ f(x)=\frac{1}{(1+x)^{2}}, \quad n=5 $$

Compute the third degree Taylor polynomial generated by \(\sin x\) at \(x=\frac{\pi}{4}\).

Suppose that \(a_{k}=f(k)\) for \(k=1,2,3, \ldots\), where \(f(x)\) is positive, decreasing, and continuous on \([1, \infty)\). Put the following expressions in order, from smallest to largest. Explain your reasoning with a picture or two. \(\sum_{k=2}^{n-1} a_{k}, \quad \sum_{k=3}^{n} a_{k}, \quad \int_{2}^{n} f(x) d x\)

Find the Maclaurin series for \(\ln (2+x)\) along with its radius of convergence.

Use your knowledge of the binomial series to find the \(n\) th degree Taylor polynomial for \(f(x)\) about \(x=0 .\) Give the radius of convergence of the corresponding Maclaurin series. One of these "series" converges for all \(x\). $$ f(x)=2(9-x)^{\frac{1}{2}}, \quad n=3 $$

Find the fth degree Taylor polynomial for \(\sqrt{x}\) centered at \(x=9\).

Explain why the hypothesis that \(f(x)\) is decreasing is important in the Integral Test.

(a) Find the Maclaurin series for \(\ln \left(\frac{1+x}{1-x}\right)\) by subtracting the Maclaurin series for \(\ln (1-x)\) from that for \(\ln (1+x)\) (b) Show that when \(x=\frac{1}{3},\left(\frac{1+x}{1-x}\right)=2\). (c) Use the rst four nonzero terms of the series in part (a) to approximate \(\ln 2 .\) Compare your answer with the approximation given by the rst four terms of the series for \(\ln (1+x)\) evaluated at \(x=1\), and the value of \(\ln 2\) given by a calculator or computer.

Use your knowledge of the binomial series to find the \(n\) th degree Taylor polynomial for \(f(x)\) about \(x=0 .\) Give the radius of convergence of the corresponding Maclaurin series. One of these "series" converges for all \(x\). $$ f(x)=\frac{x}{\sqrt{4+x}}, \quad n=3 $$

Write the third degree Taylor polynomial centered about \(x=0\) for \(f(x)=\frac{1}{(1+x)^{p}}\) where \(p\) is constant.

Use your knowledge of improper integrals to give an upper and lower bound for \(\sum_{k=1}^{\infty} \frac{1}{k^{2}}\)

Show that \(\sum_{k=0}^{\infty} \frac{(2 x)^{x}}{k !}\) is a solution to the differential equation \(f^{\prime}(x)=2 f(x) .\) What familiar function does this series represent?

(a) Expand \(f(x)=(a+x)^{4}\) by multiplying out or by using Pascal's triangle. (b) Rewrite \(f(x)\) as \(\left[a\left(1+\frac{x}{a}\right)\right]^{4}=a^{4}\left(1+\frac{x}{a}\right)^{4}\). Use the binomial series to expand \(\left(1+\frac{x}{a}\right)^{4}\), multiply by \(a^{4}\), and demonstrate that the result is the same as in part (a).

Let \(\sum_{k=1}^{\infty} a_{k}\) be a series and \(S_{k}=a_{1}+a_{2}+\cdots+a_{k}\) its \(k\) th partial sum, where \(k=1,2,3, \ldots\) Let \(L\) be a constant, \(0

Show that if \(f(x)=\sum_{k=0}^{\infty} a_{k} x^{k}\) is a power series solution to \(f^{\prime}(x)=-f(x)\), then \(f(x)=\sum_{k=0}^{\infty}(-1)^{k} \frac{x^{k}}{k !} .\) What function does this series represent?

Find the Maclaurin series for \(\frac{1}{1+x^{2}}\). What is the radius of convergence?

Introduction to Error Analysis: Let \(f(x)=e^{x}\) and let \(P_{k}(x)\) be its \(k\) th degree Taylor polynomial about \(x=0\). Graph \(R_{k}(x)=f(x)-P_{k}(x)\) for \(k=1,2, \ldots, 5 .\)

Let \(\sum_{k=1}^{\infty} a_{k}\) be a series and \(S_{n}=\sum_{k=1}^{n} a_{k}\) its \(n\) th partial sum, where \(n=1,2,3, \ldots\). For each of the following, decide whether or not enough information is given to assure that \(\sum_{k=1}^{\infty} a_{k}\) converges. \(M\) and \(m\) are constants. Explain your reasoning. (a) \(a_{k}>0\) for all \(k\) and \(S_{n}>m\) for all \(n\). (b) \(a_{k}>0\) for all \(k\) and \(S_{n}m\) for all \(n\). (d) \(m

Use power series to solve the differential equation \(f^{\prime \prime}(x)=9 f(x)\). What familiar function(s) does this series represent?

Use the binomial series to find the Maclaurin series for \(\frac{1}{\sqrt{1-x^{2}}} .\) What is the radius of convergence?

Use a third degree Taylor polynomial to approximate \(\ln 0.9\).

In Problems 25 through 32, determine whether the series converges or diverges. In this set of problems knowledge of the Limit Comparison Test is assumed. \(\sum_{k=1}^{\infty} \frac{3}{2^{k}-1}\)

The Bessel function \(J_{0}(x)\) is given by \(J_{0}(x)=\sum_{k=0}^{\infty}(-1)^{k} \frac{x^{2 k}}{(k !)^{2} 2^{2 k}} .\) It converges for all \(x\). (a) If the rst three nonzero terms of the series are used to approximate \(J_{0}(0.1)\), will the approximation be too large, or too small? Give an upper bound for the magnitude of the error. (b) How many nonzero terms of the series for \(J_{0}(1)\) must be used to approximate \(J_{0}(1)\) with error less than \(10^{-4}\) ?

Use any method to find the Maclaurin series for \(f(x) .\) (Strive for efficiency.) Determine the radius of convergence. $$ f(x)=x e^{-x} $$