MathDB

Consider

0<a<T

D=\mathbb{R}\backslash \{ kT+a\mid k\in \mathbb{Z}\}

, and let

f:D\to \mathbb{R}

T-

periodic and differentiable function which satisfies

f' > 1

(0, a)

and

f(0)=0,\lim_{\substack{x\to a\\x<a}}f(x)=+\infty \text{ and }\lim_{\substack{x\to a\\ x<a}}\frac{f'(x)}{f^2(x)}=1.

[*]Prove that for every

n\in \mathbb{N}^*

, the equation

f(x)=x

has a unique solution in the interval

(nT, nT+a)

, denoted

x_n

.[/*] [*]Let

y_n=nT+a-x_n

and

z_n=\int_0^{y_n}f(x)\text{d}x

. Prove that

\lim_{n\to \infty}{y_n}=0

and study the convergence of the series

\sum_{n=1}^{\infty}{y_n}

and

\sum_{n=1}^{n}{z_n}

Problem 4