Consider the series [infinity]
∑ (-1)^n (x+1)^n/ (n+1)!
n=0

a. Find the series' radius and interval of convergence.
b. For what values of x does the series converge absolutely?
c. For what values of x does the series converge coditionally?

(a) If

then by the ratio test, the series converges for all x, since

so the series radius of convergence is ∞ and the interval of convergence is (-∞, ∞).

(b) The series converges everywhere absolutely, because the ratio test for

also shows the radius of convergence is ∞.

(c) The series converges absolutely, so conditional convergence is moot.