## Reccurent function sequence SEEMOUS 2010

Suppose is a continuous function, and define the sequence in the following way:

.

a) Prove that the series converges for any .

b) Find an explicit formula in terms of for the above series.

*Seemous 2010, Problem 1*

a) It is obvious that . The function is continuous defined on a compact set, therefore is bounded. This means we can find such that on . But then

on .

Continuing we get

on .

By induction we have

on .

We are needed to prove that the series is convergent. For this it is enough to prove that it is absolutely convergent, and for that we have

.

This proves by using the Weierstrass test that the series converges (moreover it converges uniformly).

b) In the second part denote by , and note that by differentiating term by term (this is possible due to the uniform convergence) we obtain

,

a first order linear differential equation which can be immediatley solved.