Power Series: find the radius of convergence R of the series

Question

n=1 x^n/6n-1

R=

find the interval, I, of convergence of the series. ( use interval notation)

I=

Answer 1

Use the Ratio Test

Let a_n = xⁿ / (6n - 1)

l a_n+1 / a_n l = l (xⁿ⁺¹ / (6n + 5) [(6n - 1) / xⁿ l = [(6n - 1) / (6n + 5)] l x l

Taking the limit of the expression above as n approaches infinity, we get (1) l x l = l x l.

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

At the endpoints, the limit is equal to one, so the Ratio Test is inconclusive.

When x = 1, a_n = 1 / (6n - 1). ∑a_n diverges (compare with the Harmonic Series).

When x = -1, ∑a_n is a convergent alternating series.

Interval of convergence is [-1, 1).