Chapter 12

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

Expand \(f(x)\) in powers of \(x\) $$f(x)=\left(x^{2}+x\right) \ln (1+x)$$

Find the interval of convergence. $$\sum \frac{7^{k}}{k !} x^{k}$$

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

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

Write the given sums as \(\sum_{k=3}^{10} a_{k}\) and as \(\sum_{i=0}^{7} a_{i+3}\) $$\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\dots+\frac{1}{17}$$

Show that the series diverges. $$\sum_{k=0}^{\infty} \frac{(-5)^{k}}{4^{k+1}}$$

Determine whether the series converges or diverges. $$\sum \frac{1}{\sqrt{k^{3}-1}}$$

Determine whether the series converges or diverse. $$\sum \frac{k^{4}-1}{3 k^{2}+5}$$

Show that the series diverges. $$\sum_{k=1}^{\infty}\left(\frac{k+1}{k}\right)^{k}$$

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

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

Test these series for (a) absolute convergence, (b) conditional convergence. \(\sum(-1)^{k} \frac{4^{k-2}}{e^{k}}\).

Transform the first expression into the second by a change of indices. $$\sum_{k=2}^{10} \frac{k}{k^{2}+1} ; \quad \sum_{n=-1}^{7} \frac{n+3}{n^{2}+6 n+10}$$

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

Show that the series diverges. $$\sum_{k=2}^{\infty} \frac{k^{k-2}}{3 k}$$

Expand \(f(x)\) in powers of \(x\) $$f(x)=x^{5}(\sin x+\cos 2 x)$$

Assume that \(f\) is a function with \(\left|f^{(n)}(x)\right| \leq 3\) for all \(n\) and all real \(x\). Find the least integer \(n\) for which you can be sure that \(P_{n}(2)\) approximates \(f(2)\) with three decimal place accuracy.

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

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

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

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

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

Transform the first expression into the second by a change of indices. $$\sum_{n=2}^{12} \frac{(-1)^{n}}{n-1} ; \quad \sum_{i=1}^{11} \frac{(-1)^{k+1}}{k}$$

Evaluate the limit (i) by using L'Hôpital's rule, (ii) by using power series. $$\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}$$

Find the interval of convergence. $$\sum(-1)^{k} \frac{2^{k}}{3^{k+1}} x^{k}$$

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

Transform the first expression into the second by a change of indices. $$\sum_{k=4}^{25} \frac{1}{k^{2}-9} ; \quad \sum_{n=7}^{28} \frac{1}{n^{2}-6 n}$$

Assume that a ball dropped to the floor rebounds to a height proportional to the height from which it was dropped. Find the total length of the path of a ball dropped from a height of 6 feet, given it rebounds initially to a height of 3 feet.

Determine whether the series converges or diverges. $$\sum k^{-(1+1 / k)}$$

Determine whether the series converges or diverse. $$\sum \frac{1+2^{k}}{1+5^{k}}$$

Evaluate the limit (i) by using L'Hôpital's rule, (ii) by using power series. $$\lim _{x \rightarrow 0} \frac{\sin x-x}{x^{2}}$$

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

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

Expand \(g(x)\) as indicated. $$g(x)=x^{2}+e^{3 x} \quad \text { in poivers of } x-2.$$

Determine whether the series converges or diverges. $$\sum \frac{11}{1+100^{-2}}$$

Determine whether the series converges or diverse. $$\sum \frac{k^{3 / 2}}{k^{5 / 2}+2 k-1}$$

How much money must you deposit at \(r \%\) interest compounded annually to enable your descendants to withdraw \(n_{1}\) dollars at the end of the first year, \(n_{2}\) dollars at the end of the second year, \(n_{3}\) dollars at the end of the third year, and so on in perpetuity? Assume that the set of \(n_{k}\) is bounded above, \(n_{k} \leq N\) for all \(k\), and express your answer as an infinite series.

Evaluate the limit (i) by using L'Hôpital's rule, (ii) by using power series. $$\lim _{x \rightarrow 0} \frac{\cos x-1}{x \sin x}$$

Find the interval of convergence. $$\sum(-1)^{k} \frac{k !}{k^{3}}(x-1)^{k}$$

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

Expand \(g(x)\) as indicated. $$g(x)=x \sin x \quad \text { in powers of } x.$$

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

Determine whether the series converges or diverse. $$\sum \frac{2 k+1}{\sqrt{k^{4}+1}}$$

Express the decimal fraction \(0 . a_{1} a_{2} \cdots a_{n}\) in sigma notation using powers of \(1 / 10\).

Evaluate the limit (i) by using L'Hôpital's rule, (ii) by using power series. $$\lim _{x \rightarrow 0} \frac{e^{x}-1-x}{x \arctan x}$$

Use Taylor polynomials to estimate the following within 0.01. $$\sin 0.3$$

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

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

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