What is the interval of convergence of sum_1^oo ((5^n)*(x-1)^n)/n?

1 Answer
George C.
Jan 15, 2016

[4/5, 6/5)

Explanation:

The harmonic series sum_(n=1)^oo 1/n is divergent, but the alternating version sum_(n=1)^oo (-1)^n/n is convergent.

If abs(5(x-1)) < 1 then sum_(n=1)^oo ((5^n)*(x-1)^n)/n will converge faster than a geometric series with common ratio less than 1.

If abs(5(x-1)) > 1 then sum_(n=1)^oo ((5^n)*(x-1)^n)/n will diverge since its tail diverges faster than a geometric series with common ratio greater than 1.

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