Question

determine the interval and the radius of convergence for the power series ∑ n=1 to infinity of x^4n/((4n)!)

Answer 1

Use the ratio test:
\[\large\begin{align*}\lim_{n\to\infty}\left|\frac{x^{4(n+1)}}{(4(n+1))!}\cdot\frac{(4n)!}{x^{4n}}\right|&=\lim_{n\to\infty}\left|\frac{x^{4n+4}}{(4n+4)!}\cdot\frac{(4n)!}{x^{4n}}\right|\\\\
&=\lim_{n\to\infty}\left|\frac{x^4(4n)!}{(4n+4)\cdots(4n+1)(4n)!}\right|\\\\
&=|x^4|\lim_{n\to\infty}\frac{1}{(4n+4)\cdots(4n+1)}\\\\
&=0\end{align*}\]
Since \(0<1\), the ratio test says the series converges for all \(x\). This means the radius of convergence is \(\infty\), and the interval is \((-\infty,\infty)\).