MathDB

Let

f:\mathbb{R}\to\mathbb{R}

be a continuous, strictly decreasing function such that

f([0,1])\subseteq[0,1]

.
[*]For all positive integers

n{}

prove that there exists a unique

a_n\in(0,1)

, solution of the equation

f(x)=x^n

. Moreover, if

(a_n){}

is the sequence defined as above, prove that

\lim_{n\to\infty}a_n=1

. [*]Suppose

f

has a continuous derivative, with

f(1)=0

and

f'(1)<0

. For any

x\in\mathbb{R}

we define

F(x)=\int_x^1f(t) \ dt.

Let

\alpha{}

be a real number. Study the convergence of the series

\sum_{n=1}^\infty F(a_n)^\alpha.

P4