Chapter 8

Determine whether the series is convergent or divergent. If it is convergent, find its sum. $$\sum_{k=1}^{\infty} \frac{k(k+2)}{(k+3)^{2}}$$

Find the Taylor series for \(f(x)\) centered at the given value of \(a\) . [Assume that \(f\) has a power series expansion. Do not show that \(R_{n}(x) \rightarrow 0.1]\) $$f(x)=1 / x, \quad a=-3$$

\(13-16=\) Approximate the sum of the series correct to four decimal places. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{6}} $$

Find the radius of convergence and interval of convergence of the series. $$\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n+1}}{(2 n+1) !}$$

\(9-32\) n Determine whether the sequence converges or diverges. If it converges, find the limit. $$a_{n}=\frac{3^{n+2}}{5^{n}}$$

Find a power series representation for the function and determine the radius of convergence. $$ f(x)=\ln (5-x) $$

(a) Approximate \(f\) by a Taylor polynomial with degree \(n\) at the number \(a\) . (b) Use Taylor's Formula to estimate the accuracy of the approximation \(f(x) \approx T_{n}(x)\) when \(x\) lies in the given interval. (c) Check your result in part (b) by graphing \(\left|R_{n}(x)\right|\) $$f(x)=x \sin x, \quad a=0, \quad n=4, \quad-1 \leqslant x \leqslant 1$$

Determine whether the series is convergent or divergent. If it is convergent, find its sum. $$\sum_{n=1}^{\infty} \frac{n-1}{3 n-1}$$

Find the Taylor series for \(f(x)\) centered at the given value of \(a\) . [Assume that \(f\) has a power series expansion. Do not show that \(R_{n}(x) \rightarrow 0.1]\) $$f(x)=e^{2 x}, \quad a=3$$

\(13-16=\) Approximate the sum of the series correct to four decimal places. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n-1} n^{2}}{10^{n}} $$

Find the radius of convergence and interval of convergence of the series. $$\sum_{n=0}^{\infty} \frac{(x-2)^{n}}{n^{2}+1}$$

\(9-32\) n Determine whether the sequence converges or diverges. If it converges, find the limit. $$a_{n}=\frac{n^{2}}{\sqrt{n^{3}+4 n}}$$

Find a power series representation for the function and determine the radius of convergence. $$ f(x)=x^{2} \tan ^{-1}\left(x^{3}\right) $$

Determine whether the series is convergent or divergent. If it is convergent, find its sum. $$\sum_{n=1}^{\infty} \frac{1+3^{n}}{2^{n}}$$

(a) Approximate \(f\) by a Taylor polynomial with degree \(n\) at the number \(a\) . (b) Use Taylor's Formula to estimate the accuracy of the approximation \(f(x) \approx T_{n}(x)\) when \(x\) lies in the given interval. (c) Check your result in part (b) by graphing \(\left|R_{n}(x)\right|\) $$f(x)=x \ln x, \quad a=1, \quad n=3, \quad 0.5 \leqslant x \leqslant 1.5$$

Find the Taylor series for \(f(x)\) centered at the given value of \(a\) . [Assume that \(f\) has a power series expansion. Do not show that \(R_{n}(x) \rightarrow 0.1]\) $$f(x)=\sin x, \quad a=\pi / 2$$

\(13-16=\) Approximate the sum of the series correct to four decimal places. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{3^{n} n !} $$

Find the radius of convergence and interval of convergence of the series. $$\sum_{n=0}^{\infty}(-1)^{n} \frac{(x-3)^{n}}{2 n+1}$$

\(9-32\) n Determine whether the sequence converges or diverges. If it converges, find the limit. $$a_{n}=\sqrt{\frac{n+1}{9 n+1}}$$

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

Determine whether the series is convergent or divergent. If it is convergent, find its sum. $$\sum_{n=1}^{\infty} \frac{1+2^{n}}{3^{n}}$$

Find the Taylor series for \(f(x)\) centered at the given value of \(a\) . [Assume that \(f\) has a power series expansion. Do not show that \(R_{n}(x) \rightarrow 0.1]\) $$f(x)=\cos x, \quad a=\pi$$

Is the 50 th partial sum \(s_{50}\) of the alternating series \(\Sigma_{n=1}^{\infty}(-1)^{n-1} / n\) an overestimate or an underestimate of the total sum? Explain.

Find the radius of convergence and interval of convergence of the series. $$\sum_{n=1}^{\infty} \frac{n}{b^{n}}(x-a)^{n}, \quad b>0$$

\(9-32\) n Determine whether the sequence converges or diverges. If it converges, find the limit. $$a_{n}=\frac{(-1)^{n}}{2 \sqrt{n}}$$

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

Determine whether the series is convergent or divergent. If it is convergent, find its sum. $$\sum_{n=1}^{\infty} \cos \frac{1}{n}$$

Find the Taylor series for \(f(x)\) centered at the given value of \(a\) . [Assume that \(f\) has a power series expansion. Do not show that \(R_{n}(x) \rightarrow 0.1]\) $$f(x)=\sqrt{x}, \quad a=16$$

For what values of \(p\) is the following series convergent? $$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{p}}$$

Find the radius of convergence and interval of convergence of the series. $$\sum_{n=1}^{\infty} \frac{n}{4^{n}}(x+1)^{n}$$

\(9-32\) n Determine whether the sequence converges or diverges. If it converges, find the limit. $$a_{n}=\frac{(-1)^{n+1} n}{n+\sqrt{n}}$$

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

Determine whether the series is convergent or divergent. If it is convergent, find its sum. $$\sum_{n=1}^{\infty} \sqrt[n]{2}$$

\(19-40=\) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty} \frac{n}{5^{n}} $$

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

\(9-32\) n Determine whether the sequence converges or diverges. If it converges, find the limit. $$a_{n}=\cos (n / 2)$$

Use Taylor's Formula to determine the number of terms of the Maclaurin series for \(e^{x}\) that should be used to estimate \(e^{0.1}\) to within \(0.00001 .\)

Find a power series representation for the function and determine the radius of convergence. $$ f(x)=\frac{x^{2}+x}{(1-x)^{3}} $$

Determine whether the series is convergent or divergent. If it is convergent, find its sum. $$\sum_{n=1}^{\infty}\left[(0.8)^{n-1}-(0.3)^{n}\right]$$

Suppose you know that $$f^{(n)}(4)=\frac{(-1)^{n} n !}{3^{n}(n+1)}$$ and the Taylor series of \(f\) centered at 4 converges to \(f(x)\) for all \(x\) in the interval of convergence. Show that the fifth-degree Taylor polynomial approximates \(f(5)\) with error less than \(0.0002 .\)

\(19-40=\) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty} \frac{(-2)^{n}}{n^{2}} $$

Find the radius of convergence and interval of convergence of the series. $$\sum_{n=1}^{\infty} \frac{(2 x-1)^{n}}{5^{n} \sqrt{n}}$$

\(9-32\) n Determine whether the sequence converges or diverges. If it converges, find the limit. $$a_{n}=\cos (2 / n)$$

Determine whether the series is convergent or divergent. If it is convergent, find its sum. $$\sum_{n=1}^{\infty} \arctan n$$

\(19-40=\) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=0}^{\infty} \frac{(-10)^{n}}{n !} $$

Find the radius of convergence and interval of convergence of the series. $$\sum_{n=1}^{\infty} \frac{x^{n}}{1 \cdot 3 \cdot 5 \cdot \cdot \cdots \cdot(2 n-1)}$$

\(9-32\) n Determine whether the sequence converges or diverges. If it converges, find the limit. $$\left\\{\frac{(2 n-1) !}{(2 n+1) !}\right\\}$$

Find a power series representation for \(f,\) and graph \(f\) and several partial sums \(s_{a}(x)\) on the same screen. What happens as \(n\) increases? $$ f(x)=\frac{x}{x^{2}+16} $$

Use the Alternating Series Estimation Theorem or Taylor's Formula to estimate the range of values of for which the given approximation is accurate to within the stated error. Check your answer graphically. $$\cos x \approx 1-\frac{x^{2}}{2}+\frac{x^{4}}{24} \quad( | \text { error } |<0.005)$$

Determine whether the series is convergent or divergent. If it is convergent, find its sum. $$\sum_{k=1}^{\infty}(\cos 1)^{k}$$