how to find the open interval of convergence and test the endpoints for absolute and conditonal convergence.?

Question

Thanks for any help on these. Any help is gratefully thanked in advance!

Sigma, infinite, n=1, ((x-2)^n)/(2^n)*(n^2)

Sigma, infinite, n=1, ((x+3)^n)/n!

Answer 1

use the ratio test, or by substitution of x = ... you can reduce the problem to a known serie.

a(n) = ((x+3)^n)/n!
a(n+1) = ((x+3)^[n+1])/n+1!
a(n+1)/a(n) = (x+3)/(n+1)
lim n-+INF | (x+3)/(n+1) | = 0. R=0 , thus convergence for any x.