## Definite parametric integral

Let . Evaluate the integral

**Solution**

The key substitution is . Applying it we see that

Thus,

Thus , .

## Power of matrix

Let . Prove that .

**Solution**

The characteristic polynomial of is . This in return means and . Thus,

## Gamma infinite product

Prove that

**Solution**

Converting the product to a sum and using duplication formula for the gamma function and telescoping,

Using Stirling formula

we get that

## Upper bound of max product

Let be the roots of the polynomial

Prove that:

**Solution**

Suppose the roots of polynomial are where

Let . Then, the are the zeros of in the disk where is chosen such that for .

Jensen’s inequality implies that

Applying Cauchy – Schwartz yields,

Therefore,

Letting and we get the result.

## Harmonic limit

Let denote the – th harmonic number. Evaluate the limit

**Solution**

**Lemma: **It holds that .

*Proof: *We have successively:

Thus,