There are two identities that have a seemingly dual correspondence:

ex=∑n≥0xnn!

and

n!=∫∞0xnex dx.

Is there anything to this comparison? (I vaguely remember a generating function/integration correspondence)

Are there similar sum/integration pairs for other well-known (or not-so-well-known) functions?

**Answer**

There is a close relationship between the two identities, but I don’t know if the exact formal similarity is anything other than a neat coincidence along the lines of the Sophomore’s dream (although I could of course be wrong about this). First note that the second identity can be written as

1=∫∞0e−xxnn!dx

and therefore it is equivalent to the identity

11−t=∞∑n=0tn∫∞0e−xxnn!dx=∫∞0e−xetxdx.

which is an application of the first identity. (This new identity is easy to prove, since the integrand is just e(t−1)x so it has antiderivative 1t−1e(t−1)x and the identity follows from here.)

I know of interesting explanations of the two identities separately which are somewhat related, but not another direct connection like the one above: for the first see this math.SE question and for the second see this math.SE question.

**Attribution***Source : Link , Question Author : Mitch , Answer Author : Community*