Question

Show that the series is convergent or divergent:

I need the b) part so badly

Answer 1

`\displaystyle\sum_(n\ge0)(2e^n)/(e^(2n)+1)=\sum_(n\ge0)\frac2{e^n+\frac1{e^n}}`

As
`n\to\infty`,
`\frac1{e^n}` becomes insignificant, so you can compare this series to


`\displaystyle\sum_(n\ge0)\frac2{e^n}`

since


`e^n<e^n+\frac1{e^n}\implies\frac2{e^n}>\frac2{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=(2e)/(e-1)`

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