Chapter 12

Find the Taylor polynomial \(P_{5}\) for the given function \(f\). $$f(x)=\tan x$$

Test these series for (a) absolute convergence, (b) conditional convergence. \(\sum\left(\frac{1}{k}-\frac{1}{k !}\right)\).

Evaluate. $$\sum_{k=2}^{4} \frac{1}{3^{k-1}}$$

Expand \(f(x)\) in powers of \(x,\) basing your calculations of the tangent series. $$\tan x=x+\frac{1}{3} x^{3}+\frac{2}{15} x^{5}+\frac{17}{315} x^{7}+\cdots$$ $$f(x)=\sec ^{2} x$$

Expand \(g(x)\) as indicated and specify the values of \(x\) for which the expansion is valid. \(g(x)=3 x^{3}-2 x^{2}+4 x+1 \quad\) in powers of \(x-1.\)

Determine whether the series converges or diverges. $$\sum \frac{k^{2}+2}{k^{3}+6 k}$$

Determine whether the series converges or diverse. $$\sum \frac{1}{\sqrt{2 k^{2}-k}}$$

Expand \(f(x)\) in powers of \(x,\) basing your calculations of the tangent series. $$\tan x=x+\frac{1}{3} x^{3}+\frac{2}{15} x^{5}+\frac{17}{315} x^{7}+\cdots$$ $$f(x)=\ln \cos x$$

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

Find the Taylor polynomial \(P_{5}\) for the given function \(f\). $$f(x): x \cos x^{2}$$

Test these series for (a) absolute convergence, (b) conditional convergence. \(\sum \frac{k^{3}}{2^{k}}\).

Find the sum of the series. $$\sum_{k=0}^{\infty} \frac{1}{2^{k+3}}$$

Determine whether the series converges or diverse. $$\sum\left(\frac{2}{5}\right)^{k}$$

Evaluate. $$\sum_{k=1}^{5} \frac{(-1)^{2}}{k !}$$

Determine whether the series converges or diverges. $$\sum \frac{1}{(\ln k)^{k}}$$

Find the interval of convergence. $$\sum \frac{1}{k 2^{k}} x^{k}$$

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

Find \(f^{(9)}(0)\) $$f(x)=x^{2} \sin x$$

Expand \(g(x)\) as indicated and specify the values of \(x\) for which the expansion is valid. \(g(x)=2 x^{5}+x^{2}-3 x-5 \quad\) in powers of \(x+1\).

Determine \(P_{0}(x), P_{1}(x), P_{2}(x), P_{3}(x)\) for $$ f(x)=1-x+3 x^{2} \cdot 5 x^{3} $$

Test these series for (a) absolute convergence, (b) conditional convergence. \(\sum(-1)^{k}-\frac{1}{2 k+1}\).

Find the sum of the series. $$\sum_{k=0}^{\infty} \frac{2^{k+3}}{3^{k}}$$

Determine whether the series converges or diverges. $$\sum \frac{1}{(\ln k)^{10}}$$

Test these series for (a) absolute convergence, (b) conditional convergence. \(\sum(-1)^{k} \frac{(k !)^{2}}{(2 k) !}\).

Evaluate. $$\sum_{k=0}^{3}(-1)^{k}\left(\frac{1}{2}\right)^{2 k}$$

Find the interval of convergence. $$\sum \frac{1}{k^{2} 2^{k}} x^{k}$$

Expand \(g(x)\) as indicated and specify the values of \(x\) for which the expansion is valid. \(g(x)=x^{-1} \quad\) in powers of \(x-1\).

Find the sum of the series. $$\sum_{k=0}^{\infty} \frac{3^{k-1}}{4^{3 k+1}}$$

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

Expand \(f(x)\) in powers of $x$$$f(x)=\sin x^{2}$$

Find the interval of convergence. $$\sum\left(\frac{k}{100}\right)^{k} x^{k}$$

Use series to show that every repeating decimal fraction represents a rational number (the quotient of two integers).

Express in sigma notation. $$1+3+5+7 \cdots \cdots+21$$

Determine the \(n\) th Taylor polynomial \(P_{n}\) for the function \(f\). $$f(x)=e^{-x}$$

Determine whether the series converges or diverges. $$\sum \frac{1}{1+\sqrt{k}}$$

Test these series for (a) absolute convergence, (b) conditional convergence. \(\sum \frac{k !}{(-2)^{k}}\).

Determine whether the series converges or diverse. $$\sum \frac{1}{k^{2 / 3}}$$

Find the interval of convergence. $$\sum\left(\frac{k}{100}\right)^{k} x^{k}$$

Determine whether the series converges or diverse. $$\sum \frac{1}{k(k+1)(k+2)}$$

Express in sigma notation. $$1-3+5-7+\dots-19$$

Determine the \(n\) th Taylor polynomial \(P_{n}\) for the function \(f\). $$f(x)=\sinh x$$

Expand \(g(x)\) as indicated and specify the values of \(x\) for which the expansion is valid. \(g(x)=(b+x)^{-1} \quad\) in powers of \(x-a, \quad a \neq-b\)

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

Test these series for (a) absolute convergence, (b) conditional convergence. \(\sum \sin \left(\frac{k \pi}{4}\right)\).

Expand \(f(x)\) in powers of \(x\) $$f(x)=e^{3 x^{3}}$$

Derive the indicated result by appealing to the geometric series. $$\sum_{k=0}^{\infty}(-1)^{k} x^{k}=\frac{1}{1+x}, \quad|x|<1$$

Find the interval of convergence. $$\sum \frac{2^{k}}{\sqrt{k}} x^{k}$$

Expand \(g(x)\) as indicated and specify the values of \(x\) for which the expansion is valid. \(g(x)=(1-2 x)^{-1} \quad\) in powers of \(x+2\).

Determine whether the series converges or diverges. $$\sum \frac{k !}{10^{4 k}}$$

Test these series for (a) absolute convergence, (b) conditional convergence. \(\Sigma(-1)^{2}(\sqrt{k-1}-\sqrt{k})\).