A very tough convergence: (-1)^n * [e-(1+1/n)^n]

[emnethesemn]

Hi and sorry if I misplaced the thread.
I'm having quite some trouble with analyzing the convergence of the following series :

$$\sum$$ (-1)^n * [e-(1+1/n)^n]

I had troubles both with absolute and normal convergence.

With normal convergence
I tried Leibniz
1)
lim a(n) = 0 Which is ok { lim e-(1+1/n)^n = 0 } as
(1+1/n)^n rises to e when we let n go to infinity.
2)
a(n)=<v>=a(n+1)
I get to a part in which I have [1+1/n]/[1+1/(n+1)]<=1 I put limes on both sides
and get them to equalize. So I guess normal convergence is fine. But I'm not sure of this.



When testing absolute convergence I figured out that I could state that it's smaller then
$$\sum$$|e|+|(1+1/n)^n|
but these series diverge so I'm nowhere.
I tried integral test but an integral of [(1+1/n)^n]dn with range of 0 to +infinity doesn't seem solvable.

I need to prove it Diverges absolutely, any ideas?
Sorry for all the bad grammar and anything that I left unclear, it's 1:40am and I just couldn't get LATEX to work, any guides on that matter would be appreciated as well.

Thanks in advance.
Just noticed, this is a self study and I guess I posted it in the wrong place as I assumed. My apologies.

 

[mathman]

Hi and sorry if I misplaced the thread.
I'm having quite some trouble with analyzing the convergence of the following series :

$$\sum$$ (-1)^n * [e-(1+1/n)^n]

I had troubles both with absolute and normal convergence.

With normal convergence
I tried Leibniz
1)
lim a(n) = 0 Which is ok { lim e-(1+1/n)^n = 0 } as
(1+1/n)^n rises to e when we let n go to infinity.
2)
a(n)=<v>=a(n+1)
I get to a part in which I have [1+1/n]/[1+1/(n+1)]<=1 I put limes on both sides
and get them to equalize. So I guess normal convergence is fine. But I'm not sure of this.



When testing absolute convergence I figured out that I could state that it's smaller then
$$\sum$$|e|+|(1+1/n)^n|
but these series diverge so I'm nowhere.
I tried integral test but an integral of [(1+1/n)^n]dn with range of 0 to +infinity doesn't seem solvable.

I need to prove it Diverges absolutely, any ideas?
Sorry for all the bad grammar and anything that I left unclear, it's 1:40am and I just couldn't get LATEX to work, any guides on that matter would be appreciated as well.

Thanks in advance.
Just noticed, this is a self study and I guess I posted it in the wrong place as I assumed. My apologies.

For absolute convergence work with:
$$\sum$$|e-(1+1/n)^n|

 

[Gib Z]

I don't think what you did with the alternating series is correct, and in any case you are over complicating the issue. $\sum (-1)^n a_n$ converges if and only if a) $a_n \to 0$ and b) a_n is eventually decreasing. You already said those 2 facts for this sum in that first section you called "1)", you were done.

For absolute convergence, try showing $$| a_n | = e\left( \frac{1}{2n} + O\left(\frac{1}{n^2}\right) \right)$$.