> Why do you think it's correct?!

It's not - you gave a good example. If it could calculate a few more terms it would get the correct result.
If it could try n=4, then correct result would've been evaluated.

> How general this workaround is?

I don't think this is the best solution. Best solution would be implementing asymptotic series.
For now I thought this should work but you gave a good example where it would not.
I would try and test the result in a way such that it tries further values of n

What do you suggest?
Should we go on with this workaround for now or wait till asymptotic expansion is implemented?