# Convergence or Divergence of Factorial Series

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1. Apr 26, 2015

### Christian M.

How can I find out if 1/n! is divergent or convergent?

I cannot solve it using integral test because the expression contains a factorial.

I also tried solving it using Divergence test. The limit of 1/n! as n approaches infinity is zero. So it follows that no information can be obtained using this test.

Is there any way that I can prove its divergence or convergence?

2. Apr 26, 2015

### pasmith

3. Apr 26, 2015

### Christian M.

I used the ratio test and got zero as the final answer. So, this means that the given factorial series is convergent.

Just for a follow-up question, is it true then that all factorial series are convergent?

4. Apr 26, 2015

### mathman

If you are asking about any series summing reciprocals of factorials, the answer is yes as long as they are all different, since any such series is bounded by the sum of all of them (which = e).

5. Apr 26, 2015

### WWGD

You can use either ratio test or Taylor series for $e^x$ at $x=1$, as proposed already , or the comparison of $1/n!$ with $1/n^2$, noticing that for n>3, $n!>n^2$

6. May 14, 2015

### phion

Take a look at $$\Sigma\frac {n^n}{n!}$$.

7. May 14, 2015

### mathman

Obviously not. The sequence $\frac{n^n}{n!}$ itself diverges, so there is no way you could sum it.