(Okay, I'll confess: I knew the first term, and I got Maple to calculate the others just now.)

So I found myself wondering -- why this n

^{-3/2}? Let D

_{n}= 4

^{-n}C

_{n}. Then and so we get ; furthermore that sum is about -(3/2) log n, for large n, and so D

_{n}is about n

_{-3/2}. The replacement of 1-x with exp(-x) obviously would need to be justified (and such justification would explain the presence of the mysterious π) but I'm still amused that this simple computation got the exponent.

## 3 comments:

Can you use Stirling's formula to get the asymptotic expression?

Yes, Stirling's formula will work. (More generally, I know how to derive this series without using Maple, and could post about it, but the original post was something I quickly dashed off before going to a talk and so I didn't want to get into details that would likely be incorrect.)

Indeed, Stirling's Formula gets you the first term, including the root pi, effortlessly.

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