Proof that series diverges ?

sid9221 · May 21, 2012

[tex] \sum_{1}^{\infty} (-1)^n(1+\frac{1}{n})^n [/tex]

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

Can't think of any other test...

tiny-tim · May 21, 2012

hi sid9221!

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

what does it look like?

sid9221 · May 21, 2012

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 · May 21, 2012

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 · May 21, 2012

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

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

Ray Vickson · May 21, 2012

sid9221 said: ↑

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 [itex] S = t_1 - t_2 + t_3 - t_4 + \cdots,[/itex] with all [itex] t_i > 0 \;( \text{or all } < 0),[/itex] you need [itex]|t_n| \rightarrow 0 [/itex] as [itex] n \rightarrow 0.[/itex] Of course the factors [itex] (-1)^n[/itex] do not have a limit, but that is not important.

RGV

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Proof that series diverges ?

sid9221

tiny-tim

sid9221

Mark44

Staff: Mentor

sid9221

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