Question

Find the interval and radius of convergence for the series. show all work and be sure to include expansions for each of your claims. Series of (-2)^n x^n / n^1/4

Answer 1

\[\large\sum_{n=1}^\infty \frac{(-2x)^n}{n^{1/4}}\]
Use the ratio test:
\[\large\begin{align*}\lim_{n\to\infty}\left|\frac{(-2x)^{n+1}}{(n+1)^{1/4}}\cdot\frac{n^{1/4}}{(-2x)^n}\right|&=\lim_{n\to\infty}\left|-2x\frac{n^{1/4}}{(n+1)^{1/4}}\right|\\
&=2|x|\lim_{n\to\infty}\frac{n^{1/4}}{(n+1)^{1/4}}\\
&=2|x|\end{align*}\]
The series converges by the ratio test if the limit is less than 1, so the series must satisfy \(2|x|<1\) to converge, or \(|x|<\dfrac{1}{2}\).