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I was asked to find the value of the following summation.

$$ x=1+\sum_{i=1}^{\infty}\frac{1}{2^i}+\sum_{j=1}^{\infty}\frac{1}{3^j}+\sum_{k=1}^{\infty}\frac{1}{5^k} $$

I "solved" it approximately by expanding a few of the terms for each of the summations and finding the fraction that each of them appears to converge on, which resulted in

$$ x=1+1+\frac{1}{2}+\frac{1}{4}=\frac{11}{4} $$

But I am told that this is not correct. Even if it were, my method feels shaky. What is a more solid mathematical way to do this?

EDIT: After having plugged my expression into WolframAlpha, it seems that my answer is correct after all. But I'm still interested in knowing if there's a better way.

stett
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  • Each of the series you have given is a geometric series, which has a well-known formula for its sum. – dannum Jan 17 '15 at 01:26

1 Answers1

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Hint: $1+x+x^2+\cdots = \dfrac{1}{1-x}$, with $|x| < 1$

DeepSea
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    It is important to note that the series above do not contain the $x^0$ term. Simply subtracting $1$ gives $$x+x^2+x^3+\dots=\frac{x}{1-x}$$ for $|x|\lt1$. – robjohn Jan 17 '15 at 02:28