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A set of interesting math problems which have simple and elegant solutions or proofs.

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Reciprocals of Integers Series

You have a series in which each element s_n = 1/(n+1). The question is: does this series converge or diverge? In other words, does it have a finite (convergent) or infinite (divergent) sum?

Solution

The series diverges. This can be seen easily by dividing it into groups as follows:

G₁ = 1/2
G₂ = 1/3 + 1/4
G₃ = 1/5 + 1/6 + 1/7 + 1/8

Each group G_k contains 2^{k - 1} elements, and each element is greater than or equal to the last element in the group. Thus, the numerical value of each group is greater than (the number of elements in the group) times (the last element in the group).

Thus, for all groups, G_k >= 2^k-1*1/2^k.

This means the value of each group is greater than or equal to 1/2. Summing an infinite number of values greater than 1/2 results in an infinite value, so the series diverges.

Isn't that slick?

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"Slick Math" is written by Douglas Twitchell, and hosted at The Problem Site.

Contents copyright 2008 by Douglas Twitchell. Contents of this page may not be reproduced without permission of the author.