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

Mathematics Asked on January 5, 2022

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.

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!

Answered by OlympusHero on January 5, 2022

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 $$x$$.

Answered by Alex on January 5, 2022