Determining the interval of convergence of Power series

Question

Lets say you have a function f(x)= (1/4)+(2/4^2)+(3/4^3)+...+(((n+1)*x^n)/4^(n+1) for all x

a) What would the interval of convergence of this power series be

b) Find the limit as x approaches 0 of [f(x)-(1/4)]/x

Answer 1

The ratio test gives the interval of convergence as |x|<4.

The limit will come from the term in n=1, i.e. 1/8.