Chapter 12

Determine whether the series converges or diverse. $$\Sigma\left(\frac{4}{3}\right)^{k}$$

Express in sigma notation. $$1 \cdot 2+2 \cdot 3+3 \cdot 4+\dots-35 \cdot 36$$

Expand \(f(x)\) in powers of \(x\) $$f(x)=\frac{1-x}{1+x}$$

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

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

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

Derive the indicated result by appealing to the geometric series. $$\sum_{i=1}^{2}(-1)^{k} x^{2 k}-\frac{1}{1-x^{2}} \cdot|x|<1$$

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

Determine the \(n\) th Taylor polynomial \(P_{n}\) for the function \(f\). $$f(x)=\ln (1-x)$$

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

Express in sigma notation. $$\text { The lower sum } m_{1} \Delta x_{1}+m_{2} \Delta x_{2}+\cdots+m_{n} \Delta x_{n}$$

Find a series expansion for the expression. $$\frac{x}{1-x} \text { for } | x,<1$$

Expand \(f(x)\) in powers of \(x\) $$f(x)=\frac{2 x}{1-x^{2}}$$

Determine the \(n\) th Taylor polynomial \(P_{n}\) for the function \(f\). $$f(x)=e^{r x}, \quad r \text { a real number. }$$

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

Express in sigma notation. $$\text { The upper } \operatorname{sum} M_{1} \Delta x_{i}+M_{2} \Delta x_{2}+\cdots+M_{n} \Delta x_{n}$$

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

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

Test these series for (a) absolute convergence, (b) conditional convergence. \(\sum \sin \left(\frac{\pi}{4 h^{2}}\right)\).

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

Find a series expansion for the expression. $$\frac{x}{1-x} \text { for } | x|<1$$

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

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

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

Determine the \(n\) th Taylor polynomial \(P_{n}\) for the function \(f\). $$f(x)=\cos b x . \quad b \text { a real number. }$$

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

Express in sigma notation. $$\text { The Rjemann sum } f\left(x_{i}^{*}\right) \Delta x_{1}+f\left(x_{2}^{*}\right) \Delta x_{2}+\cdots+f\left(x_{n}^{*}\right) \Delta x_{n}$$

Find a series expansion for the expression. $$\frac{x}{x+x^{2}} \quad \text { for }|x|<1$$

Expand \(f(x)\) in powers of \(x\) $$f(x)=\frac{1}{1-x}+e^{x}$$

Find the interval of convergence. $$\sum \frac{k}{10^{k}} x^{k}$$

Assume that \(f\) is a function with \(\left|f^{(n)}(x)\right| \leq 1\) for all \(n\) and all real \(x\). (The sine and cosine functions have this property.) Estimate the maximum possible error if \(P_{5}(1 / 2)\) is used to estimate \(f(1 / 2)\).

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

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

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

Write the given sums as \(\sum_{k=3}^{10} a_{k}\) and as \(\sum_{i=0}^{7} a_{i+3}\) $$\frac{1}{2^{3}}+\frac{1}{2^{4}}-\cdots-\frac{1}{2^{10}}$$

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

Find a series expansion for the expression. $$\frac{x}{1+4 x^{2}} \text { for }|x|<\frac{1}{2}$$

Assume that \(f\) is a function with \(\left|f^{(n)}(x)\right| \leq 1\) for all \(n\) and all real \(x\). (The sine and cosine functions have this property.) Estimate the maximum possible error if \(P_{7}(-2)\) is used to estimate \(f(-2)\)

Determine whether the series converges or diverges. $$\sum \frac{1}{k}\left(\frac{1}{\ln k}\right)^{3 / 2}$$

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

Expand \(f(x)\) in powers of \(x\) $$f(x)=\cosh x \sinh x$$

Find the interval of convergence. $$\sum \frac{3 k^{2}}{e^{k}} x^{k}$$

Determine whether the series converges or diverse. $$\sum \frac{7 k+2}{2 k^{5}+7}$$

Write the given sums as \(\sum_{k=3}^{10} a_{k}\) and as \(\sum_{i=0}^{7} a_{i+3}\) $$\frac{3^{3}}{3 !}+\frac{4^{4}}{4 !}+\cdots+\frac{10^{10}}{10 !}$$

Show that the series diverges. $$1+\frac{3}{2}+\frac{9}{4}+\frac{27}{8}+\frac{81}{16}+\dots$$

Expand \(f(x)\) in powers of \(x\) $$f(x)=x \ln \left(1+x^{3}\right)$$

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

Determine whether the series converges or diverse. $$\sum \frac{2 k+5}{5 k^{3}+3 k^{2}}$$

Write the given sums as \(\sum_{k=3}^{10} a_{k}\) and as \(\sum_{i=0}^{7} a_{i+3}\) $$\frac{3}{4}-\frac{4}{5}+\dots-\frac{10}{11}$$

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