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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$\begingroup$Yes, by taking $b_n=\frac{1}{n!}$ and use the Leibniz 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 ratio test we can find more. In fact, you series is absolutely convergent as well($\sum_{0}^{\infty}\frac{1}{n!}=\text{e}$). $$\lim_{n\to\infty}\left|\frac{b_{n+1}}{b_n}\right|=0<1$$ Note that in Leibniz test, you can also take limit of $|b_n|$ instead of $b_n$.
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