Question

# Determine the radius of convergence and the interval of convergence for each power series. sum_{n=1}^infty n!(2x-1)^n

Series
Determine the radius of convergence and the interval of convergence for each power series.
$$\sum_{n=1}^\infty n!(2x-1)^n$$

2020-12-01
The given series is:
$$\sum_{n=1}^\infty n!(2x-1)^n$$
Applying Ratio test to the above series:
$$L=\lim_{n\to\infty}|\frac{a_{n+1}}{a_n}|$$
$$=\lim_{n\to\infty}|\frac{(n+1)!(2x-1)^{n+1}}{n!(2x-1)^n}|$$
$$=\lim_{n\to\infty}|\frac{(n+1)!(2x-1)^{n}(2x-1)}{n!(2x-1)^n}|$$
$$=\lim_{n\to\infty}|(n+1)(2x-1)|$$
$$=|2x-1|\lim_{n\to\infty}(n+1)$$
This means that $$L=\infty$$ provided $$x\ne\frac{1}{2}$$
Hence, the given series only converges when $$x=\frac{1}{2}$$
Therefore, the radius of convergence = 0 and interval of convergence is $$x=\frac{1}{2}$$