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I have come across the series:

$\sum_{j=1}^\infty \frac{x^j (j-1)}{j! \sqrt{2j - 1}}$

which is easily seen to be absolutely convergent everywhere (e.g. ratio test). It seems that it should be very close to $\exp(x)$ and I would like to characterize it exactly in terms of simple functions if possible. Does anyone have ideas?

Cheers.

  • 0
    I can now show at least the first limit which is all I need for my purposes: $\frac{\exp(-x)}{x} \sum_{j=1}^\infty \frac{x^j (j-1)}{j! \sqrt{2j - 1}} \rightarrow 0$ as $x \rightarrow \infty$. Note that for $j \geq 1$, $\frac{j-1}{\sqrt{2j - 1}} \leq \sqrt{j}$. So $\frac{\exp(-x)}{x} \sum_{j=1}^\infty \frac{x^j (j-1)}{j! \sqrt{2j - 1}} \leq \frac{1}{x} \mathbb{E}[\sqrt{Y}]$ where $Y$ is a Poisson random variable with mean $x$. By Jensen's inequality this is bounded above by $\sqrt{x} / x = 1 / \sqrt{x}$ and the limit follows.2012-08-29

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