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I want to prove if $\sum\limits_{n=2}^\infty (-1)^n \frac{1}{n!}$ is convergent or not.

I can prove this by the $ratio \ test$.

Isn't this series an alternating series with $a_n = (-1)^n$ and $b_n = \frac{1}{n!}$?

If it is, I can prove with the $Leibniz \ test$ because $b_n>= b_{n+1}$ and $lim_{n\to \infty} b_n = 0$

Is this correct?

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Yes. You did it correctly. –  B. S. Jan 4 '13 at 19:30
    
Yes, series $\sum\limits_{n=2}^\infty (-1)^n \frac{1}{n!}$ converges by Leibniz test. –  M. Strochyk Jan 4 '13 at 19:34

1 Answer 1

up vote 3 down vote accepted

Yes, by taking $b_n=\frac{1}{n!}$ and use the Libnitz test you see that $$\lim_{n\to\infty}b_n=0$$ and $$b_n=\frac{1}{n!}\geq\frac{1}{(n+1)!}=b_{n+1}, n\geq1$$ so the series converges, however using the raio test we can find more. In fact, you series is absolutely convergent as well($\sum_{0}^{\infty}\frac{1}{n!}=\text{e}$). $$\lim_{n\to\infty}\big|\frac{b_{n+1}}{b_n}|=0<1$$ Note that in Liebnitz test, you can also take limit of $|b_n|$ instead of $b_n$.

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Thanks for your explanation –  Favolas Jan 4 '13 at 19:52
    
@Favolas: I am glad that I could help you. ;-) –  B. S. Jan 4 '13 at 19:53
    
Greeeeeeeeaaaat! +1 –  amWhy Feb 24 '13 at 0:06

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