Power Series

Find the radius and the open interval of convergence of the following power series.

(a) \(\displaystyle{\sum_{n=1}^{\infty}(-1)^n\,\frac{x^n}{n^3}}\)   solution

(b) \(\displaystyle{\sum_{n=1}^{\infty}\frac{(x-5)^n}{n}}\)   solution

(c) \(\displaystyle{\sum_{n=1}^{\infty}(-1)^n \frac{ n^2\,x^n}{2^n}}\)  solution

(d) \(\displaystyle{\sum_{n=1}^{\infty} \frac{ (x+2)^n}{4^n}}\)   solution

(e) \(\displaystyle{\sum_{n=1}^{\infty}\frac{(x+2)^n}{n^2\,3^n}}\)  solution

(f) \(\displaystyle{\sum_{n=1}^{\infty} (-2)^n \frac{x^n}{3\sqrt{n}}}\)  solution

(g) \(\displaystyle{\sum_{n=1}^\infty \frac{(-3)^n }{n}x^n} \)  solution