13: If you tried the first approach, you should get

Now it is necessary to integrate, this is a typical case for the integration by parts.

If you used partial fractions, you should get

Note that the factoring out of x⁻¹ worked only if x is not 0. What is the sum of this series when x = 0?

If you used the third approach, you should get

By the way, note that the given series also converges for x = 1, which is a number where none of the above procedures work. One can try to find this sum directly. Use the telescopic series approach to do it.

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