Question

Show that the series is convergent or divergent: I need the b) part so badly

Answer 1

$\displaystyle\sum_{n\ge0}\frac{2e^n}{e^{2n}+1}=\sum_{n\ge0}\frac2{e^n+\frac1{e^n}}$

As $n\to\infty$, $\dfrac1{e^n}$ becomes insignificant, so you can compare this series to

$\displaystyle\sum_{n\ge0}\frac2{e^n}$

since

$e^n\dfrac2{e^n+\frac1{e^n}}$

This comparison series converges as it's geometric:

$\displaystyle\sum_{n\ge0}\frac2{e^n}=2\sum_{n\ge0}\left(\frac1e\right)^n=\frac{2e}{e-1}$

Since this convergent series is smaller than the first series, the first series must also be convergent.