compute $\prod_{n=2}^{N} \frac{n(n+1) +1}{ n(n-1)+1}$ .

Question

I am not able to compute this infinite product and so I am asking for help here.

Compute the product $\prod_{n=2}^{N} \frac{n(n+1) +1}{ n(n-1)+1}$ .

I tried by factoring $\frac{n(n+1) +1}{ n(n-1)+1}$ into $(1+ \frac{2n}{n^2-n+1}) $ but this doesn't seems to help and I don't have any other ideas.

Please help.

Answer 1

That is a telescopic product, for $ N\geq 2 $, we have : \begin{aligned} \prod_{n=2}^{N}{\frac{n^{2}+n+1}{n^{2}-n+1}}=\prod_{n=2}^{N}{\frac{\left(n+1\right)^{2}-\left(n+1\right)+1}{n^{2}-n+1}}&=\frac{\left(N+1\right)^{2}-\left(N+1\right)+1}{2^{2}-2+1}\\ &= \frac{N^{2}+N+1}{3}\end{aligned}

A telescopic product, has the form of $ \prod\limits_{k=p}^{q}{\frac{a_{k+1}}{a_{k}}}=\frac{a_{q+1}}{a_{p}} $, were $ p,q\in\mathbb{N} $, such that $ q\geq p $, and $ \left(a_{k}\right)\in\left(\mathbb{R}^{*}\right)^{\mathbb{N}} $.