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# A series with least common multiple

Let be a strictly increasing sequence of positive numbers. For all denote as the least common multiple of the first terms of the sequence. Prove that , as , the following sum converges

Solution

This is a result due to Paul Erdös stating that if are natural numbers such that then

and the original question follows since the sum we seek is less or equal to .

However, we are presenting another proof. Denote as the average order of the numbers , i.e.,

For any we have where is the product of primes not present in the factorization of . Note that are squarefree integers. Note also that it may be an empty product, i.e., . Then

It is easy to see (and show by induction) that so we have

Hence, Consequently, we have

So the sum of reciprocals of converges. Then, by Cesàro summation, we see that

also converges.

## 1 Comment

1. John Khan says:

Probably a familiar series for testing of convergence is:

Let be a a strictly increasing sequence of positive integers. Prove that the series converges where denotes the least common multiple.

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