Question

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

Answer 1

`[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`.