Searching for an asymptotic to exp[0.5]

[jaydfox]

...and comparing your power series, it easily follows that your power series is asymptotic to exp(x) x^(1/2). Hmm, oh dear, I think you missed a +1 in the Gamma function in your series? (LOL, it's okay, I usually forget the +1 as well.) If you set beta=-1/2, then all is well!

Haha, I think my math is wrong somewhere, because I double-checked in Excel, and you were right! I obviously did something wrong in my math, though I haven't figured out what yet... I'll update this post when I get it sorted out, LOL!

Update: I think I figured it out. I was doing the Gamma function correctly (beta = -1/2), but I was extracting beta from the exponent incorrectly. I had my equations backwards. So my original equation (with the beta's intact) is still correct... I think...

\(
\frac{\exp(x)}{\sqrt{x}} \approx \sum_{k=0}^{\infty}\frac{x^{k}}{\Gamma(k+3/2)}
\)

\(
\exp(x)\sqrt{x} \approx \sum_{k=0}^{\infty}\frac{x^{k}}{\Gamma(k+1/2)}
\)

And generally:
\(
\exp(x) x^{-\beta} \approx \sum_{k=0}^{\infty}\frac{x^{k}}{\Gamma(k+\beta+1)}
\)