What is the real number called to which the sequence $$\gamma_n =1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{n} - \log _e n$$ converges and what is the radius of convergence?
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1The Euler-Mascheroni constant. – angryavian Aug 21 '14 at 15:44
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4Radius of convergence is irrelevant here, since it is not a power series. – angryavian Aug 21 '14 at 15:45
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This is the Euler-Mascheroni constant, $\gamma \approx 0.5772 \ldots$
Ayesha
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In addition to this answer, it is not know whether or not $\gamma$ can be expressed as a rational number. – pshmath0 Aug 21 '14 at 16:20
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$$\lim_{n\to\infty}\sum^n_{k=1}\frac{1}{k}-\ln(n) = \gamma$$ It is widely known as the Euler–Mascheroni constant which is defined as the limiting difference between the harmonic series and the natural logarithm.
$$\gamma \approx 0.5772\space1566\space4901\space5328\space6060\space6512\space090(1)$$
The radius of curvature is not valid for this series as it not a power-series.
Nick
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