Does the Series \(\sum_{n=1}^{\infty} (-1)^n(1+\frac{1}{n})^n\) Diverge?

[sid9221]

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

I've tried the alternating series test but the "b_n" part converges to e.

Can't think of any other test...

 

[tiny-tim]

hi sid9221!

hint: draw a_n against n on graph paper (as dots) …

what does it look like? ​

 

[sid9221]

I know I've seen the plot somewhere (can't remember the technical name).

It looks like a sin function but one that's increasing .

I know it diverges just can't prove it.

 

[Mark44]

Use the "n-th term test for divergence." If the limit of the n-th term in the series is different from 0 or doesn't exist, the series diverges.

 

[sid9221]

How to you take the limit of (-1)^n.

Wolfram alpha says e^2i 0 to Pi ?!

 

[Ray Vickson]

How to you take the limit of (-1)^n.

Wolfram alpha says e^2i 0 to Pi ?!

You are missing the whole point. If you have a series $S = t_1 - t_2 + t_3 - t_4 + \cdots,$ with all $t_i > 0 \;( \text{or all } < 0),$ you need $|t_n| \rightarrow 0$ as $n \rightarrow 0.$ Of course the factors $(-1)^n$ do not have a limit, but that is not important.

RGV