Timeline for Asymptotic expansion of $\sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$
Current License: CC BY-SA 3.0
12 events
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| Apr 3, 2017 at 19:27 | comment | added | Johannes Trost | @j.s. no, I think it was not the transition from sum to integral. I guess, it is the complicated expansion in both $n$ and $x$. In the approach in this answer I had to expand the exponent of an exponential, therefore include logarithmic terms in a clean way, which I could not figure out correctly. In the approach of my other answer, the distinction is made quite clear in the asymptotic expansion of the hypergeometric, where the exponentially suppressed terms are easily identified. But this was done already in formula of Volkmer and Wood, that I cited, so I did not have to take care. | |
| Apr 3, 2017 at 18:25 | comment | added | j.c. | Do you know where this approach broke down exactly? In the approximation of the sum by an integral? | |
| Apr 2, 2017 at 20:30 | history | edited | Johannes Trost | CC BY-SA 3.0 |
Added note at the beginning to warn the reader.
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| Apr 2, 2017 at 13:18 | comment | added | esg | @Johannes Trost: no offence, and thanks for confirming the expansions | |
| Mar 31, 2017 at 13:34 | comment | added | Todd Trimble | @esg Would you be able to expand your comment into an answer? Johannes indicates here, at meta.mathoverflow.net/questions/3190/…, that he thinks his answer should not be the accepted one, and that he would look favorably on your posting an answer based on your findings. | |
| Mar 31, 2017 at 12:21 | history | edited | Johannes Trost | CC BY-SA 3.0 |
The coefficients of L(x) are wrong as well. esg's comment gives the correct numbers.
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| Mar 31, 2017 at 12:15 | comment | added | Johannes Trost | I have strong numerical evidence that at least the coefficients of $x^{-2}$ and $x^{-4}$ in the expansions given for $C(x)$ and $L(x)$ are correct (and thus mine are false). | |
| Mar 29, 2017 at 16:20 | comment | added | esg | Using probabilistic reasoning I get \begin{align*} C(x)\,e^{-x^2}&= 1+\frac{3}{8}x^{-2} + \frac{65}{128}x^{-4} + \frac{1225}{1024}x^{-6} + \frac{1619583}{425984}x^{-8}+{O}(x^{-10})\\ L(x)\,x^2\,e^{-x^2}&= 1+\frac{15}{8}x^{-2} + \frac{665}{128}x^{-4}+\frac{19845}{1024}x^{-6}+\frac{37475823 }{425984}x^{-8}+{O}(x^{-10}) \end{align*} These differ from your first findings but seem to be close to your later findings (I haven't checked the numerics). | |
| Mar 28, 2017 at 11:43 | history | edited | Johannes Trost | CC BY-SA 3.0 |
Edited answer on evidence for the asymptotic expansion to be numerically very inaccurate.
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| Mar 19, 2017 at 21:42 | history | edited | Johannes Trost | CC BY-SA 3.0 |
There was a wrong value of coefficient in first $n_{0}$. Corrected.
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| Mar 19, 2017 at 18:20 | vote | accept | Trax | ||
| Apr 21, 2017 at 22:27 | |||||
| Mar 18, 2017 at 22:42 | history | answered | Johannes Trost | CC BY-SA 3.0 |