Chapter 8

If the radius of convergence of the power series \(\Sigma_{n-0}^{\infty} c_{n} x^{n}\) is \(10,\) what is the radius of convergence of the series \(\sum_{n=1}^{\infty} n c_{n} x^{n-1} ?\) Why?

(a) Find the Taylor polynomials up to degree 6 for \(f(x)=\cos x\) centered at \(a=0 .\) Graph \(f\) and these polynomials on a common screen. (b) Evaluate \(f\) and these polynomials at \(x=\pi / 4, \pi / 2,\) and \(\pi .\) (c) Comment on how the Taylor polynomials converge to \(f(x) .\)

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

(a) What is an alternating series? (b) Under what conditions does an alternating series converge? (c) If these conditions are satisfied, what can you say about the remainder after \(n\) terms?

Draw a picture to show that $$\sum_{n=2}^{\infty} \frac{1}{n^{1.3}}<\int_{1}^{\infty} \frac{1}{x^{1.3}} d x$$ What can you conclude about the series?

(a) What is a sequence? (b) What does it mean to say that \(\lim _{n \rightarrow \infty} a_{n}=8 ?\) (c) What does it mean to say that \(\lim _{n \rightarrow \infty} a_{n}=\infty\) ?

Suppose you know that the scrics \(\Sigma_{n=0}^{\infty} b_{n} x^{n}\) converges for \(|x|<2 .\) What can you say about the following series? Why? $$ \sum_{n=0}^{\infty} \frac{b_{n}}{n+1} x^{n+1} $$

(a) Find the Taylor polynomials up to degree 3 for \(f(x)=1 / x\) centered at \(a=1 .\) Graph \(f\) and these polynomials on a common screen. (b) Evaluate \(f\) and these polynomials at \(x=0.9\) and 1.3 (c) Comment on how the Taylor polynomials converge to \(f(x) .\)

Explain what it means to say that \(\Sigma_{n=1}^{\infty} a_{n}=5\)

The graph of \(f\) is shown. $$\begin{array}{l}{\text { (a) Explain why the serics }} \\\ {1.6-0.8(x-1)+0.4(x-1)^{2}-0.1(x-1)^{3}+\cdots} \\ {\text { is not the Taylor series of } f \text { centered at } 1 \text { . }}\end{array}$$ $$\begin{array}{l}{\text { (b) Explain why the series }} \\\ {2.8+0.5(x-2)+1.5(x-2)^{2}-0.1(x-2)^{3}+\cdots} \\ {\text { is not the Taylor series of } f \text { centered at } 2 \text { . }}\end{array}$$

What can you say about the series \(\Sigma a_{n}\) in each of the following cases? (a) $$\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=8$$ (b) $$\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=0.8$$ (c) $$\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=1$$

Suppose \(f\) is a continuous positive decreasing function for \(x \geqslant 1\) and \(a_{n}=f(n) .\) By drawing a picture, rank the following three quantities in increasing order: $$\int_{1}^{6} f(x) d x \quad \sum_{i=1}^{5} a_{i} \quad \sum_{i=2}^{6} a_{i}$$

(a) What is the radius of convergence of a power series? How do you find it? (b) What is the interval of convergence of a power series? How do you find it?

(a) What is a convergent sequence? Give two examples. (b) What is a divergent sequence? Give two examples.

Find a power series representation for the function and determine the interval of convergence. $$ f(x)=\frac{1}{1+x} $$

Find the Taylor polynomial \(T_{3}(x)\) for the function \(f\) centered at the number \(a .\) Graph \(f\) and \(T_{3}\) on the same screen. $$f(x)=1 / x, \quad a=2$$

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? $$\sum_{n=1}^{\infty} \frac{1}{n^{3}}$$

\(\begin{array}{l}{\text { If } f^{(n)}(0)=(n+1) ! \text { for } n=0,1,2, \ldots, \text { find the Maclaurin }} \\ {\text { series for } f \text { and its radius of convergence. }}\end{array}\)

\(3-8=\) Test the series for convergence or divergence. $$ \frac{4}{7}-\frac{4}{8}+\frac{4}{9}-\frac{4}{10}+\frac{4}{11}-\cdots $$

Find the radius of convergence and interval of convergence of the series. $$\sum_{n=1}^{\infty}(-1)^{n} n x^{n}$$

Suppose \(\Sigma a_{n}\) and \(\Sigma b_{n}\) are series with positive terms and \(\Sigma b_{n}\) is known to be convergent. (a) If \(a_{n}>b_{n}\) for all \(n,\) what can you say about \(\Sigma a_{n} ?\) Why? (b) If \(a_{n}

List the first six terms of the sequence defined by $$a_{n}=\frac{n}{2 n+1}$$ Does the sequence appear to have a limit? If so, find it.

Find a power series representation for the function and determine the interval of convergence. $$ f(x)=\frac{5}{1-4 x^{2}} $$

Find the Taylor polynomial \(T_{3}(x)\) for the function \(f\) centered at the number \(a .\) Graph \(f\) and \(T_{3}\) on the same screen. $$f(x)=e^{-x} \sin x, \quad a=0$$

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? $$\sum_{n=1}^{\infty} \frac{1}{\ln (n+1)}$$

\(3-8=\) Test the series for convergence or divergence. $$ -\frac{3}{4}+\frac{5}{5}-\frac{7}{6}+\frac{9}{7}-\frac{11}{8}+\cdots $$

Find the radius of convergence and interval of convergence of the series. $$\sum_{n=1}^{\infty} \frac{(-1)^{n} x^{n}}{\sqrt[3]{n}}$$

Suppose \(\Sigma a_{n}\) and \(\Sigma b_{n}\) are series with positive terms and \(\sum b_{n}\) is known to be divergent. (a) If \(a_{n}>b_{n}\) for all \(n,\) what can you say about \(\Sigma a_{n} ?\) Why? (b) If \(a_{n}

List the first nine terms of the sequence \(\\{\cos (n \pi / 3)\\} .\) Does this sequence appear to have a limit? If so, find it. If not, explain why.

Find the Taylor series for \(\int\) centered at 4 if $$f^{(n)}(4)=\frac{(-1)^{n} n !}{3^{n}(n+1)}$$ What is the radius of convergence of the Taylor series?

Find a power series representation for the function and determine the interval of convergence. $$ f(x)=\frac{2}{3-x} $$

Find the Taylor polynomial \(T_{3}(x)\) for the function \(f\) centered at the number \(a .\) Graph \(f\) and \(T_{3}\) on the same screen. $$f(x)=\cos x, \quad a=\pi / 2$$

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? $$\sum_{n=1}^{\infty} \frac{n}{1+\sqrt{n}}$$

Find the Maclaurin series for \(f(x)\) using the definition of a Maclaurin series. [Assume that \(f\) has a power series expansion. Do not show that \(R_{n}(x) \rightarrow 0 . ]\) Also find the associated radius of convergence. $$f(x)=(1-x)^{-2}$$

\(3-8=\) Test the series for convergence or divergence. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{2 n+1} $$

Find the radius of convergence and interval of convergence of the series. $$\sum_{n=1}^{\infty} \frac{x^{n}}{2 n-1}$$

It is important to distinguish between $$\sum_{n=1}^{\infty} n^{b} \quad \text { and } \quad \sum_{n=1}^{\infty} b^{n}$$ What name is given to the first series? To the second? For what values of \(b\) does the first series converge? For what values of \(b\) does the second series converge?

\(5-8=\) Find a formula for the general term \(a_{n}\) of the sequence, assuming that the pattern of the first few terms continues. $$\left\\{-3,2,-\frac{4}{3}, \frac{8}{9},-\frac{16}{27}, \ldots\right\\}$$

Find a power series representation for the function and determine the interval of convergence. $$ f(x)=\frac{1}{x+10} $$

Find the Taylor polynomial \(T_{3}(x)\) for the function \(f\) centered at the number \(a .\) Graph \(f\) and \(T_{3}\) on the same screen. $$f(x)=\frac{\ln x}{x}, \quad a=1$$

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? $$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n !}$$

Find the Maclaurin series for \(f(x)\) using the definition of a Maclaurin series. [Assume that \(f\) has a power series expansion. Do not show that \(R_{n}(x) \rightarrow 0 . ]\) Also find the associated radius of convergence. $$f(x)=e^{-2 x}$$

\(3-8=\) Test the series for convergence or divergence. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{n}{\sqrt{n^{3}+2}} $$

Find the radius of convergence and interval of convergence of the series. $$\sum_{n=1}^{\infty} \frac{(-1)^{n} x^{n}}{n^{2}}$$

Use the Integral Test to determine whether the series is convergent or divergent. $$\sum_{n=1}^{\infty} \frac{1}{n^{5}}$$

\(5-8=\) Find a formula for the general term \(a_{n}\) of the sequence, assuming that the pattern of the first few terms continues. $$\left\\{1,-\frac{1}{3}, \frac{1}{9},-\frac{1}{27}, \frac{1}{81}, \ldots\right\\}$$

Find a power series representation for the function and determine the interval of convergence. $$ f(x)=\frac{x}{9+x^{2}} $$

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

Find the Taylor polynomial \(T_{3}(x)\) for the function \(f\) centered at the number \(a .\) Graph \(f\) and \(T_{3}\) on the same screen. $$f(x)=x e^{-2 x}, \quad a=0$$

Find the Maclaurin series for \(f(x)\) using the definition of a Maclaurin series. [Assume that \(f\) has a power series expansion. Do not show that \(R_{n}(x) \rightarrow 0 . ]\) Also find the associated radius of convergence. $$f(x)=\sin \pi x$$