Summation and radius of convergence help!!?

Question

E from n=0 to infinity of [(-1)^n (x-4)^n] / (n+1)

We need a radius of convergence

Answer 1

If you take absolute value
Ix-4I^n /(n+1) .If you apply the nth root criterium the limit for n=>infinity is Ix-4I
If Ix-4I<1 3 At the extreme +5 you have the serie (-1)^n *1/(n+1) which is the
alternate harmonic serie (convergent)

At rhe extreme 3 you have (-1)^n* (-1)^n /(n+1)= 1/(n+1) w hich is the harmonic serie ( divergent)

For x>5 or x<-5 the serie is divergent

Answer 2

E from n=0 to infinity of [(-1)^n (x-4)^n] / (n+1)
=E from n=0 to infinity of [(4-x)^n] / (n+1)

radius of convergence is r = 1 or |x-4|<1

Answer 3

prepare the Ratio attempt. a(n+a million)/a(n) = 2^(n+a million) x^(n+a million) n! /(n+a million)! 2^n x^n =2x/(n+a million) As n procedures infinity, 2/(n+a million) procedures 0 The series converges if |x| < a million The radius of convergence is a million.