Chapter 30

Use the Ratio Test or Root Test to find the radius of convergence of the power series given. \(\sum_{k=3}^{\infty} \frac{(x-1)^{2 k}}{(k-1) !}\)

Use the Ratio Test or Root Test to find the radius of convergence of the power series given. \(\sum_{n=1}^{\infty} \frac{(x+2)^{n}}{n(2 n+3)}\)

In Problems 60 through 71 , find the interval of convergence of the series. Explain your reasoning fully. \(\sum_{k=1}^{\infty} \frac{(3 x)^{k}}{2^{k}}\)

Find the interval of convergence of the series. Explain your reasoning fully. \(\sum_{k=1}^{\infty} 2^{k}(x-3)^{k}\)

Find the interval of convergence of the series. Explain your reasoning fully. \(\sum_{k=1}^{\infty}(-1)^{k} \frac{(x+1)^{k}}{3 k}\)

Find the interval of convergence of the series. Explain your reasoning fully. \(\sum_{k=1}^{\infty}(-1)^{k} \frac{(x+1)^{k}}{k}\)

Find the interval of convergence of the series. Explain your reasoning fully. \(\sum_{k=2}^{\infty} \frac{(x-3)^{k}}{2^{k}}\)

Find the interval of convergence of the series. Explain your reasoning fully. \(\sum_{k=1}^{\infty} \frac{(x-3)^{k}}{k !}\)

Find the interval of convergence of the series. Explain your reasoning fully. \(\sum_{k=1}^{\infty} \frac{(x-2)^{k}}{k 5^{k}}\)

Find the interval of convergence of the series. Explain your reasoning fully. \(\sum_{k=1}^{\infty} \frac{(x-1)^{k}}{k^{5}}\)

Find the interval of convergence of the series. Explain your reasoning fully. \(\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{n}}{(2 n+1)}\)

Find the interval of convergence of the series. Explain your reasoning fully. \(\sum_{k=1}^{\infty} \frac{2^{k} x^{k}}{k !}\)

Suppose \(\lim _{k \rightarrow \infty} \sqrt[k]{\left|a_{k}\right|}=\frac{1}{3}\). (a) What is the radius of convergence of \(\sum_{k=1}^{\infty} a_{k}(x-1)^{k}\) ? (b) For each value of \(x\) listed below, determine whether the series converges absolutely, converges conditionally, or diverges. i. \(x=0\) ii. \(x=-3\) iii. \(x=-1.5\) iv. \(x=5\)

Give an example of each of the following. (a) a series that converges only at \(x=4\) (b) a series that converges for \(x \in(3,5)\) and diverges otherwise (c) a series that converges for all \(x\) (d) a series that converges for \(x \in(2,6)\) and diverges otherwise

Show that if a power series \(\sum_{k=0}^{\infty} a_{k} x^{k}\) has radius of convergence \(R\), then \(\sum_{k=0}^{\infty} a_{k}(x-b)^{k}\) also has a radius of convergence of \(R\).

If \(R\) is the radius of convergence of \(\sum_{k=0}^{\infty} a_{k} x^{k}\), determine the radius of convergence of the following. (a) \(\sum_{k=100}^{\infty} a_{k} x^{k}\) (b) \(\sum_{k=0}^{\infty} a_{k}(2 x)^{k}\) (c) \(\sum_{k=0}^{\infty} a_{k}\left(\frac{x}{2}\right)^{k}\)

Show that the radius of convergence of the binomial series is 1 .