Gamma time!

$\large \gamma = \lim_{x\to1^+} \sum_{n=1}^\infty \left( \dfrac1{n^x} - \dfrac1{x^n} \right)$

Let $\gamma$ denote the Euler-Mascheroni constant. Prove the limit above.

This is a part of the set Formidable Series and Integrals

#Calculus

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Comments

$\displaystyle \lim_{x\to1^+} \sum_{n=1}^\infty \left( \dfrac1{n^x} - \dfrac1{x^n} \right) = \lim_{x\to1^+} \sum_{n=1}^\infty \left( \dfrac1{n^x} - \int_{n}^{n+1} \dfrac{\mathrm{d}y}{y^x} \right)$

$\displaystyle = \sum_{n=1}^\infty \left( \dfrac1{n} - \int_{n}^{n+1} \dfrac{\mathrm{d}y}{y} \right)$

$\displaystyle= \lim_{n \to \infty} \left( H_{n} - \log n \right)$

$= \boxed{\gamma}$

Ishan Singh - 5 years, 3 months ago

exact intended solution!

upvoted :)

Hamza A - 5 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$