Mathematics

Problem

Consider the power series

\displaystyle\sum_{n=0}^{\infty} \dfrac{k^{\,n}}{n!}\,x^{n}

, where

k

is a fixed positive constant. Which statement best describes the effect of changing

k

on the radius of convergence of the series? a) The radius equals

1/k

; increasing

k

decreases the radius. b) The radius equals

0

; the series converges only at

x=0

.},{ c) The radius is infinite and therefore independent of

k

. d) The radius equals

\sqrt{k}

; increasing

k

increases the radius but not proportionally.