- E12
[0D6]Topics:Euler-Mascheroni constant.Prerequisites:[211].
Show that the limit
\[ πΎ = \lim _{n β β } \left( β_{k=1}^ n \frac 1{k} - \log ( n) \right)\quad . \]exists and is finite. This \(πΎ\) is called Costante di Eulero - Mascheroni. It can be defined in many different ways (see the previous link) including
\[ πΎ = β«_ 1^β\left(\frac{1}{β xβ}-\frac{1}{x}\right)\, {\mathbb {d}}x \]where the parentheses \(β β β\) indicate the floor function \(β x β{\stackrel{.}{=}}\max \{ nββ€ :nβ€ x\} \). In the image 1 the constant \(πΎ\) is the blue area.
![\includegraphics[width=0.4\textwidth ]{UUID/0/D/7/blob_zxx}](/UUID/EDB/0D6/html/images/img-0001.png)
Figure 1 Representation of Euler-Mascheroni constant Image by William Demchick, Creative Commons Attribution 3.0 Unported License, taken from wikipedia.
Solution 1
EDB β 0D6
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English
Authors:
"Mennucci , Andrea C. G."
.
Bibliography
Book index
Book index
- convergence, of a series
- Euler-Mascheroni constant
- constant, Euler-Mascheroni
- floor
- integer part
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