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\centerline{\eins Application to Euler's Constant}
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Put
\quad\quad{$\displaystyle f(x) := {\sum\limits_{n=0}^{\infty} \frac{x^n}{{n!}^2}}$}\hfill\break
\hphantom{Put}\quad\quad{$\displaystyle g(x) := {\sum\limits_{n=0}^{\infty} \left( \frac{1}{1} + \cdots + \frac{1}{n} \right) \frac{x^n}{{n!}^2}}$}\hfill\break
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Then
\hfill{$\displaystyle \frac{g(x)}{f(x)} = \frac{1}{2} \log x + \gamma + O \left( e^{-4 \sqrt{x}} \right) $.}\hfill\break
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Choose $\displaystyle {x = {\left\lceil \frac{N \log 2}{4} \right\rceil}^2}$.

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