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## “Upper” bound

Let be a function satisfying

for all positive real numbers and . Prove that

Solution

For starters, let us assume that . Dividing the interval into subintervals each of length so that . Thus,

The inequality implies that

Hence,

The limit exists and equals to . Hence , the inequality is proved for .

Now, assume that . Dividing the interval into subintervals each of length so that . Thus,

The inequality implies that

Hence,

The limit exists and equals to . Hence , the inequality is also proved for . This completes the proof!

## Sum over all positive rationals

For a rational number that equals in lowest terms , let . Prove that:

Solution

First of all we note that

Moreover for we have that

Hence for we have that

## Bessel series

Let denote the Bessel function of the first kind. Prove that:

Solution

The Jacobi – Anger expansion tells us that

Hence by Parseval’s Theorem it follows that:

## Pell – Lucas series

The Pell – Lucas numbers are defined as follows and for every it holds that

Prove that

## Determinant

Let and  such that . Prove that

Solution

Note that

Since is real, its complex eigenvalues come in conjugate pairs. Thus, in this case we conclude that has eigenvalues .

Now, if is an eigenvalue of , then is an eigenvalue of . Thus, the matrix has eigenvalues and .

Now, is the product of these eigenvalues, which is to say

as desired.

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