Series Convergence: Exploring the Limit and Root Test Methods

[azatkgz]

Homework Statement

Determine whether the series converges or diverges.


$$\sum_{n=1}^{\infty}\left(e-\left(1+\frac{1}{n}\right)^n\right)^p$$ where p is a parameter[/tex]

The Attempt at a Solution

$$\lim_{n\rightarrow\infty}e-\left(1+\frac{1}{n}\right)^n=0$$

so by using Root Test i decided that

$$\limsup_{n\rightarrow\infty}\sqrt[n]{\left(e-\left(1+\frac{1}{n}\right)^n\right)^p}<1$$
Which gives that series converges

 

[morphism]

What if p=0, for example?

 

[azatkgz]

Oh,I see.Limit is 1.I must try another way.

 

[azatkgz]

$$\sum_{n=1}^{\infty}\left(e-\left(1+\frac{1}{n}\right)^n\right)^p=\sum_{n=1}^{\infty}\left(e-e^{n\ln \left(1+\frac{1}{n}\right)}\right)^p=\sum_{n=1}^{\infty}\left(e-e^{n\left(\frac{1}{n}-\frac{1}{2n^2}+O(\frac{1}{n^3})\right)}\right)^p=\sum_{n=1}^{\infty}\left(e-e^{1-\frac{1}{2n}+O(\frac{1}{n^2})}\right)^p$$

$$=\sum_{n=1}^{\infty}e^p\left(1-e^{-\frac{1}{2n}+O(\frac{1}{n^2})}\right)^p=\sum_{n=1}^{\infty}e^p\left(1-(1-\frac{1}{2n}+O(\frac{1}{n^2}))\right)^p$$

$$=\sum_{n=1}^{\infty}e^p\left(\frac{1}{2n}+O(\frac{1}{n^2})\right)^p$$

Is it converges for all p<1?