Limit of a Sum of a sequence

[K1ngbadlex]

Hello, I hope everybody is fine, I need help for the limit of a Sum of a sequence :
Thank you!

 

[Dr.Peterson]

I think Arithmetic is the wrong category for this ...

Please tell us, as we ask you to, the context of your question (what course you are taking, what you have learned that might be relevant) and where you are stuck in your work. If you haven't thought of anything to try, what have you thought of that you didn't bother trying?

 

[pka]

Hello, I hope everybody is fine, I need help for the limit of a Sum of a sequence : View attachment 13780

To K1ngbadlex, I will give you some hints that do not answer give the answer.
Rather I expect you to tell us the solution. You can help others by doing so.
If \(\displaystyle |x|<1\) then \(\displaystyle S(x) = \sum\limits_{k = 1}^\infty {{x^{k - 1}}} = \frac{1}{{1 - x}}\)
The function \(\displaystyle S(x)\) is integrable thus \(\displaystyle \int {S(x)} = \sum\limits_{k = 1}^\infty {\frac{{{x^k}}}{k}} = - \log |1 - x|\).
Now what must \(\displaystyle x=\;\;\;\) to give us your question?