# Infinite Product

• Nov 23rd 2009, 12:35 AM
dymin3
Infinite Product
1. Find the value of Π(from n=1 to infinity) (1 + 1/n^2).
I know that I need to use the product expansion for sin
πz but i do not know what to do (Worried)

2. Suppose that p is a polynomial of degree n and that |p(z)|
≤ M if |z| = 1. Show that |p(z)| ≥ M|z|^n if |z|≥ 1.

I don't know how to solve this T-T

Plz help me!!!

• Nov 23rd 2009, 04:17 AM
chisigma
The 'infinite product' of $\frac{\sin x}{x}$ is...

$\frac{\sin x}{x} = \prod_{n=1}^{\infty} (1-\frac{x^{2}}{\pi^{2}\cdot n^{2}})$ (1)

... and from (1) we derive...

$\frac{\sin ix}{ix}= \frac{\sinh x}{x} = \prod_{n=1}^{\infty} (1+\frac{x^{2}}{\pi^{2}\cdot n^{2}})$ (2)

Setting in (2) $x=\pi$ we obtain...

$\prod_{n=1}^{\infty} (1+\frac{1}{n^{2}}) = \frac{\sinh \pi}{\pi}$ (3)

Kind regards

$\chi$ $\sigma$
• Nov 23rd 2009, 05:10 AM
chisigma
Quote:

Originally Posted by dymin3
2. Suppose that p is a polynomial of degree n and that |p(z)| ≤ M if |z| = 1. Show that |p(z)| ≥ M|z|^n if |z|≥ 1.

If $n>0$ [i.e. the polynom ial is not a constant...], that is a particular case of Liouville's theorem...

http://en.wikipedia.org/wiki/Liouvil...mplex_analysis)

Kind regards

$\chi$ $\sigma$
• Nov 23rd 2009, 01:06 PM
dymin3
Thank you so much!!! Have a great day!!!