# 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 ## Add your own answers! ## Related Questions ### Why is the silhouette coefficient of a clustering between$[-1,+1]$? 1 Asked on February 13, 2021 by theobands ### What is the measure of the set of sequences with convergent subsequences in the space of all real sequences? 1 Asked on February 13, 2021 by adam-rubinson ### Possible mistake in the solution of Baby Rudin Ch. 6 Ex. 11 1 Asked on February 13, 2021 ### Relation between gradient norm and distance to optimizer 0 Asked on February 13, 2021 by durdi ### Clarification about proof involving product/sum manipulation 1 Asked on February 13, 2021 by user859358 ### When can we interchange the derivative with an expectation? 2 Asked on February 13, 2021 ### When “If$A$is true then$B$is true”, is it valid to assert that “If$B$is false,$A$must also be false”? 3 Asked on February 13, 2021 by user3801230 ### proof for$eleq 3(v-2)$[Why does$d(f) geq 3$?] 1 Asked on February 13, 2021 ### Proof of prime numbers 1 Asked on February 13, 2021 by lorenza-fuller ### Laurent series example 1 Asked on February 13, 2021 by exzone ### Partial derivative holding something constant 1 Asked on February 13, 2021 by lss ### Is this claim about orthogonal projection true? 0 Asked on February 13, 2021 ### Why is the realification of a simple complex Lie algebra a semisimple real Lie algebra? 2 Asked on February 13, 2021 by richardfatman ### Is it “$Rightarrow$” or “$Leftarrow$” in “$x^2 = 9 text{ ? } x = 3\$”?

3  Asked on February 12, 2021 by physor

### Why in Cantor’s Intersection Theorem the infinite intersection must contain only one point?

1  Asked on February 12, 2021 by krishan

### How to test the series for convergence involving repeating sequences in the denominator and numerator?

2  Asked on February 12, 2021 by floppo

### Find intersection measure of a probability space

0  Asked on February 12, 2021 by eod-enaj

4  Asked on February 12, 2021 by sal-cognato

### Is the matrix exponential independent of the norm?

1  Asked on February 12, 2021 by user736690

### Kernel of Jordan Normal Form

1  Asked on February 12, 2021 by smirniy007