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.

Skip to content
# Joy of Mathematics

## Fibonacci series

## A telescopic (!) series

## Telescopic Fibonacci product

## A complex inequality

## An exponential series

A site of university mathematics

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:

Let denote the Fibonacci sequence. Prove that

**Solution**

Let denote the partial product. Thus,

Let such that and . Prove that .

**Solution**

We have successively:

Prove that

**Solution**

We recall the product identity

Thus,