Power series: Find the radius and 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)
\(\displaystyle{\sum_{n=0}^{\infty}\frac{3^n\,x^n}{2n^3+10}}\)
solution
(f) \(\displaystyle{\sum_{n=1}^{\infty}\frac{(x+2)^n}{n^2\,3^n}}\)
solution
(g) \(\displaystyle{\sum_{n=1}^{\infty}n^n\, x^n}\)
solution
(h) \(\displaystyle{\sum_{n=0}^{\infty} (-1)^n
\frac{x^{2n+1}}{(2n+1)!} }\)
solution
Find the radius and the open interval of convergence of
the following power series. (The case when only the open interval is asked.)
(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