infinite series math

Question

how to find interval of convergence of the power series:-

summation of n from 1 to infinity of ((-1)ⁿ(x-1)ⁿ)/(n^1/2)?

Answer 1

Use the Ratio Test:

l (a_n+1 / a_n) l = l x - 1 l [n / (n + 1)]^1/2

Taking the limit as n approaches infinity, we get l x - 1 l.

So, by the Ratio Test, the series converges absolutely when l x - 1 l < 1.

That is, the series converges when -1 < x - 1 < 1. So, 0 < x < 2.

Check the endpoints:

When x = 0, we have ∑_{(n = 1 to infinity)} [(-1)²ⁿ (1 / n^1/2)] = ∑_{(n = 1 to infinity)} (1/ n^1/2), which is a divergent

p-series.

When x = 2, we have ∑_{(n = 1 to infinity)} [(-1)ⁿ(1 / n^1/2)], which converges by the Alternating Series Test.

Interval of Convergence: (0, 2]. Radius of convergence = 1.