Nifty Little Math Equation

Here's a nifty little math problem: evaluate

This video shows one way to do it.

The answer is surprising; scroll down for it:

.

.

.

.

.

.

.

.

.

.

.

.
.
.
.
.
.
.
.
.
.
.
.
.
.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Added next day: Using a spreadsheet, I calculated this up to n=143 (after which nⁿ gets to large for Excel 365), when the left-hand term is 0.376 734, which differs from 1/e by 2.4%. A pinned comment below the video has a clever way to do the limit:

though the proof of Stirling's approximation is very similar to the method used in the video.