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

**Solution**

We have successively:

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# Tag: Series

## A convergent sequence

## Fibonacci series

## A telescopic (!) series

## An exponential series

## A MÃ¶bius inversion sum

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

**Solution**

We have successively:

Let denote the – th Fibonacci number. Prove that:

- .
- .
- .

**Solution**

- Taking partial sums we see that the series telescopes:
- Using Cassini’s identity we note that:
- Combining the previous results we have successively:
The result follows.

Prove that

**Solution**

We have successively:

Prove that

**Solution**

We recall the product identity

Thus,

Let such that . Prove that

**Solution**

We have successively:

since whenever