Problem

====================

Let

with the following recursion relation

Start with the index n = 1 and group to the index n = 20. How can you tell that the results are incorrect and at what index n?

Attempt

====================

I do not think that the results can be incorrect; I think it is true for all indices.

First, I integrate

using integration by parts (I forgot how to do the "evaluate from/to" symbol in Latex so I used a bracket)

Simplify to get

Since

, then

Thus,

Shouldn't this be true for all indices n? I wrote a MATLAB script that computed

at all indices and the results look fine.

Thank you for reading.