# Product Proof

1. Apr 2, 2008

### gop

1. The problem statement, all variables and given/known data

Proof that for n>2 and n is a natural number it holds that

$$\prod_{k=1}^{n}\frac{k^{2}+2}{k^{2}+1}<3$$

and
$$\prod_{k=1}^{n}\frac{k^{2}+2}{k^{2}+1}<\frac{3n}{n+1}$$

2. Relevant equations

3. The attempt at a solution

My best approach was to split the product over the fraction and then to arrive at a statement that looks like

$$\prod_{k=2}^{n}k^{2}+2<\prod_{k=1}^{n}k^{2}+1$$

I then tried to prove by induction that this statement holds but that doesn't really work. The best result I got (for n+1) is then

$$(\prod_{k=2}^{n}k^{2}+2)<(\prod_{k=1}^{n}k^{2}+1)\cdot\frac{n^{2}+2n+2}{n^{2}+2n+3}$$

But I can't do anything usefuel with that...

2. Apr 2, 2008

### Avodyne

You could try writing the product as the exponential of a sum, and then bounding the sum by an integral.