Defina$g:\Bbb{R}\setminus\Bbb{Z}\to\Bbb{R}$ por$$g(x)=\dfrac{1}{x}+\sum\limits_{n=1}^\infty\dfrac{2x}{x^2-n^2}.$ $ Muestra que$g$ es continuo.
$g$ se puede volver a escribir como$$g(x)=\dfrac{1}{x}+\sum\limits_{n=1}^\infty\dfrac{1}{n+x}-\dfrac{1}{n-x}.$$Let $ \ epsilon> 0$. Now, our job is to find a $ \ delta> 0$ such that whenever $ \ left | xy \ right | <\ delta$, $$\left|\dfrac{1}{x}-\dfrac{1}{y}+\sum\left(\dfrac{1}{n+x}-\dfrac{1}{n+y}+\dfrac{1}{n-y}-\dfrac{1}{n-x}\right)\right|<\epsilon.$$ That is; $$\left|\dfrac{y-x}{xy}+(y-x)\sum\left(\dfrac{1}{(n-y)(n-x)}-\dfrac{1}{(n+x)(n+y)}\right)\right|<\epsilon.$$ I believe I need to find the limit of that last sum, but I don't know how. Or at least I should find an upper bound $ A$ so that I can find a $ \ delta$ such that $ \ delta <\ dfrac {\ epsilon} {AB}$ where $ \ dfrac {1} {xy} <B $. ¿Que debería hacer?
