Question

The radius of convergence of the power series \[ \sum_{n=0}^{\infty} n! x^{n^2} \] is ...........

Hint:

When dealing with series involving factorials, the terms grow very quickly, often resulting in a radius of convergence of zero.

Answer 1

Correct Answer: 0.9

Step 1: Understanding the series.
The given power series is \( \sum_{n=0}^{\infty} n! x^{n^2} \). To find the radius of convergence, we apply the root test or ratio test. The root test involves examining the limit: \[ \lim_{n \to \infty} \left( \frac{1}{n!^{1/n^2}} \right). \] Since \( n! \) grows very quickly, the radius of convergence for this series is 0.