Can you answer this calculus question?

Question

Let f(x)=∑ x^n from n=0 to infinity. The radius of convergence of ∫ f(t) dt from 0 to x is

 

a) 0

b) 1

c) 2

d) ∞

e) none of these

 

Answer: B

 

Please show all your work.

Answer 1

∑ xⁿ from n=0 to N=(1-x^N)/(1-x).  For this to converge as N→∞, x^N must approach 0 or |x|must be <1

In this case f(x)=1/(1-x).  ∫₀^x f(t) dt = ∫₀^xdt/(1-t)=-ln(1-t)]₀^x=ln(1/(1-t))]₀^x which is ln(1/(1-x)), which is finite for -∞<x<1.