Convergence or Divergence of Factorial Series

[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?

 

[pasmith]

Try the ratio test.

 

[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?

 

[mathman]

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?

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).

 

[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, $n!>n^2$

 

[phion]

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

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

 

[mathman]

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

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