Thread: Integral with Recursion Relation

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.

Originally Posted by Paperwings

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.

Are any of them negative? Can such an integral be negative?

(consider the build up in errors with itterations)
.