Let .

- Prove that is strictly increasing on .
- If , solve the equation

**Solution**

- It is . However, recalling that
we conclude that is strictly increasing.

- We note that
On the other hand, and hence:

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# Joy of Mathematics

## An equation!

## A logarithm

## An identity

## A finite product

## A convergent sequence

A site of university mathematics

Let .

- Prove that is strictly increasing on .
- If , solve the equation

**Solution**

- It is . However, recalling that
we conclude that is strictly increasing.

- We note that
On the other hand, and hence:

Let . If , prove that .

If , prove that

Evaluate the product

**Solution**

We are making use of the identity

Hence,

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

**Solution**

We have successively: