- #1

broegger

- 257

- 0

I have to show that the function

[tex]f(x) = \sum_{n=1}^{\infty}\frac1{x^2+n^2}[/tex]

tends to 0 as [tex]x \rightarrow \infty[/tex], i.e. [tex]\lim_{x\rightarrow\infty}f(x) = 0[/tex]. How can I do this?

There is a hint that says I should use the inequality [tex] f(x) \leq \sum_{n=1}^N\tfrac1{x^2+n^2} + \sum_{n=N+1}^\infty\tfrac1{n^2} [/tex]. It is obvious that the first term approaches 0 as [tex]x \rightarrow \infty[/tex], but what about the second term?