Let . We define the sequence . Prove that

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# Joy of Mathematics

## A limit

## Sum of segments

## Generating function

## An eta Dedekind type product

## An integral inequality

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Let . We define the sequence . Prove that

Let be the – th harmonic number. Prove that

**Solution**

Recalling Cauchy’s product we have successively:

Prove that

**Solution**

Let and . Hence,

If is continuous and its derivative is strictly decreasing in then prove that

if it is also known that and .

**Solution**

Since is strictly decreasing then . Thus, is strictly increasing and . Therefore,

since for all .