# Proof that series diverges ?

1. May 21, 2012

### 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...

2. May 21, 2012

### tiny-tim

hi sid9221!

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

what does it look like?

3. May 21, 2012

### 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.

4. May 21, 2012

### Staff: Mentor

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.

5. May 21, 2012

### sid9221

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

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

6. May 21, 2012

### Ray Vickson

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