Question

Найдите предел выражения

Answer 1

$\frac{1}{2\cdot 7}+\frac{1}{7\cdot 12}+\dots+\frac{1}{(5n-3)(5n+2)}=\frac{1}{5}\left(\frac{7-2}{2\cdot7}+\frac{12-7}{7\cdot12}+\dots+\frac{5n+2-(5n-3)}{(5n-3)(5n+2)}\right)=$

$\frac{1}{5}\left(\frac{1}{2}-\frac{1}{7}+\frac{1}{7}-\frac{1}{12}+...+\frac{1}{5n-3}-\frac{1}{5n+2}\right)=\frac{1}{5}\cdot \left(\frac{1}{2}-\frac{1}{5n+2}\right)$

$\lim\limits_{n\to \infty}\frac{1}{5}\cdot \left(\frac{1}{2}-\frac{1}{5n+2}\right)=\frac{1}{5}\cdot \frac{1}{2}=\frac{1}{10}$