I don't agree that unless m = 4.
I would recommend not to simplify (2m)!/((m+1)m!m!) but instead to apply Stirling's formula to replace each factorial.
Show that for a very large ,
where denotes the mth catalan number.
So I did:
after some canceling.
Now and I'm stuck here... we are given Stirling's approximation formula for but I do not know how to apply that here to my final step...
Any help would be greatly appreciated!
Actually I think I am still a bit stuck. I did this: (After a bit of simplification)
But in the end the does not cancel and there is an extra
I've checked all my simplifications, they are all correct, how do I get rid of the and to match what's required?
Thanks again, actually I noticed another cool method, is this way also acceptable?
Now note as m gets large approachs e. Formally we can write,
Also note as m gets large approaches 1. Formally we can write,
So substituting these back in yields: