How find the solution to $ sum_{i=1}^{infty} left( frac{1}{2} right)^i left( frac{1}{i}right)$?

Question

I am having trouble trying to figure out the solution to the infinite series $ sum_{i=1}^{infty} left( frac{1}{2} right)^i left( frac{1}{i}right)$.
What kind of infinite series is this, and what would be the best approach to simplifying? My intuition tells me it's a converging series, but I'm lost on how to approach this problem.

OlympusHero · Answer

Let's get some terms!
We find $frac{1}{2}$, $frac{1}{8}$, $frac{1}{24}$, $frac{1}{64}$, $frac{1}{160}, $frac{1}{384}$.
These don't look too good. I kept finding some terms and added them up, it looks to converge around $frac{7}{10}$.
Wolfram alpha says 0.693147180559945309417232121458176568075500134360255254120...
Hope this helped!

Alex · Answer

Note that $frac{a^k}{k} = int_{0}^{a}x^{k-1}dx$, the interchange the sum and integral, get the (convergent) limit of the sum, and solve the integral for $x$.

How find the solution to $ sum_{i=1}^{infty} left( frac{1}{2} right)^i left( frac{1}{i}right)$?

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