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# Tag Archives: Real Analysis

## Differential equation

Let be a twice differentiable function such that

If , find an explicit formula of .

Solution

We have successively

## Polylogarithm series

Let denote the polylogarithm. Prove that

Solution

Since we have successively

## Convex function

Let be a convex function on a convex domain and a convex non-decreasing function on . Prove that the composition of is convex on .

Solution

We want to prove that for it holds that

We have:

## Fibonacci series

Let denote the Fibonacci sequence such that and . Prove that

Solution

It holds that

It follows from Cassini’s identity that

(1)

Setting back at we get

(2)

Since we get

(3)

Hence,

## Logarithmic integral

Evaluate the integral