Consider the alternating series $ \sum_{n=1}^{\infty} (-1)^n \frac{5}{n^3} x^n $.
Find the radius of convergence of it.
Answer:
$a_n=(-1)^n \frac{5}{n^3} $
Root test:
$ \lim_{n \to \infty} \sqrt[n]{|a_n|}=\lim_{n \to \infty} \sqrt[n]{\frac{5}{n^3}}=5$
Ratio test:
$ \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}|=\lim_{n \to \infty} |\frac{(n+1)^3}{n^3}|=1 $
So both limits are not equal.
But we know that if limits exists , then they will be equal.
So I am doing something wrong.
Help me
Best Answer
By roots test we have
$$ \lim_{n \to \infty} \sqrt[n]{|a_n|}=\lim_{n \to \infty} \sqrt[n]{\frac{5}{n^3}}=1$$
then the radius of convergence is $1$.
To check for the convergence we need to consider separately the cases $x=1$ and $x=-1$.