Complex Analysis: Theory and Applications · Problem 4.4.3

Elementary Functions — Howell & Matthews

Mathematics — Complex Analysis · radius-of-convergence

Problem

Find the radius of convergence of the following. ((a))

g(z) = \displaystyle\sum_{n=0}^{\infty} (-1)^n \frac{z^n}{(2n)!}

. ((b))

h(z) = \displaystyle\sum_{n=0}^{\infty} n! z^n

. ((c))

f(z) = \displaystyle\sum_{n=0}^{\infty} \left(\frac{4n^2}{2n+1} - \frac{6n^2}{3n+4}\right)^n z^n

. ((d))

g(z) = \displaystyle\sum_{n=0}^{\infty} \frac{(n!)^2}{(2n)!} z^n

. ((e))

h(z) = \displaystyle\sum_{n=0}^{\infty} (2-(-1)^n)^n z^n

. ((f))

f(z) = \displaystyle\sum_{n=0}^{\infty} \frac{n(n-1)z^n}{(3+4i)^n}

. ((g))

g(z) = \displaystyle\sum_{n=0}^{\infty} \left(\frac{3n+7}{4n+2}\right)^n z^n

. ((h))

h(z) = \displaystyle\sum_{n=0}^{\infty} \frac{2^n}{1+3^n} z^n

. ((i))

g(z) = \displaystyle\sum_{n=0}^{\infty} z^{2n}

. ((j))

g(z) = \displaystyle\sum_{n=0}^{\infty} \frac{n^n}{n!} z^n

. Hint:

\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n = e