Interval of convergence problem

Question

Find the interval of convergence for the power series.

∑n=0∞ nx^n

Give your answer using interval notation. If you need to use INF. If there is only one point in the interval of convergence, the interval notation is [a]. For example, if 0 is the only point in the interval of convergence, you would answer with [0].

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Answer 1

Use Ratio Test:

l (n+1)xⁿ⁺¹ / (nxⁿ) l = (n+1)/n l x l

Taking the limit as n goes to infinity, we have l x l

The series converges if l x l < 1. So, -1 < x < 1

Checking the endpoints:

When x = -1, we have -1 + 2 - 3 + 4 - 5 + ... (diverges)

When x = 1, we get 1 + 2 + 3 + 4 + ... (diverges)

So, the interval of convergence is (-1,1)