User:Dionyziz/Sandbox:Int

$$ \sum_{n = 1}^{\infty}(\frac{n}{n+1})^{(n^2)} $$

$$ \lim_{n\rightarrow\infty}\sqrt[n]{(\frac{n}{n+1})^{(n^2)}} = \lim_{n\rightarrow\infty}(\frac{n}{n+1})^{(n^2/n)} = \lim_{n\rightarrow\infty}(\frac{n}{n+1})^n = \lim_{n\rightarrow\infty}(\frac{1}{1+\frac{1}{n}})^n = \frac{1}{\displaystyle\lim_{n\rightarrow\infty}{(1+\frac{1}{n})}^n} = \frac{1}{e} < 1 $$

$$ \lim_{n\rightarrow\infty}(\frac{n}{n+1})^{(n^2)} = (\lim_{n\rightarrow\infty}(\frac{n}{n+1})^n)^2 = (\lim_{n\rightarrow\infty}(\frac{1}{1+\frac{1}{n}})^n)^2 = (\frac{1}{\displaystyle\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n})^2 = \frac{1}{e^2} \neq 0 $$