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,

Who is Tolaso?

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