Υπολογισμός αθροίσματος (11)
Δημοσιεύτηκε: Παρ Νοέμ 25, 2011 5:51 pm
![\displaystyle{\sum_{m=1}^{+\infty}\sum_{n=1}^{+\infty}\frac{(-1)^{n+m}}{[\sqrt{n+m}]^3}}](/forum/ext/geomar/texintegr/latexrender/pictures/d6af69cadb822b18e0ebb12abeef86f9.png "\displaystyle{\sum_{m=1}^{+\infty}\sum_{n=1}^{+\infty}\frac{(-1)^{n+m}}{[\sqrt{n+m}]^3}}")
, όπου ![[\cdot]](/forum/ext/geomar/texintegr/latexrender/pictures/0d81445b8c6d4644c626fbe5f442ebe2.png "[\cdot]")
το ακέραιο μέρος.![\displaystyle{\sum\limits_{m = 1}^{ + \infty } {\sum\limits_{n = 1}^{ + \infty } {\frac{{{{\left( { - 1} \right)}^{n + m}}}}{{{{\left\lfloor {\sqrt {n + m} } \right\rfloor }^3}}}} } = \mathop = \limits^{\left[ 1 \right]} = \sum\limits_{n = 1}^{ + \infty } {\frac{{{{\left( { - 1} \right)}^n}\left( {n - 1} \right)}}{{{{\left\lfloor {\sqrt n } \right\rfloor }^3}}}} = \sum\limits_{k = 1}^{ + \infty } {\sum\limits_{n = {k^2}}^{{{\left( {k + 1} \right)}^2} - 1} {\frac{{{{\left( { - 1} \right)}^n}\left( {n - 1} \right)}}{{{{\left\lfloor {\sqrt n } \right\rfloor }^3}}}} } = \mathop = \limits^{\left[ 2 \right]} = \sum\limits_{k = 1}^{ + \infty } {\frac{1}{{{k^3}}}\sum\limits_{n = {k^2}}^{{{\left( {k + 1} \right)}^2} - 1} {{{\left( { - 1} \right)}^n}\left( {n - 1} \right)} } = }](/forum/ext/geomar/texintegr/latexrender/pictures/3ed5bbf7d65544577e9dee0dce1cee74.png "\displaystyle{\sum\limits_{m = 1}^{ + \infty } {\sum\limits_{n = 1}^{ + \infty } {\frac{{{{\left( { - 1} \right)}^{n + m}}}}{{{{\left\lfloor {\sqrt {n + m} } \right\rfloor }^3}}}} } = \mathop = \limits^{\left[ 1 \right]} = \sum\limits_{n = 1}^{ + \infty } {\frac{{{{\left( { - 1} \right)}^n}\left( {n - 1} \right)}}{{{{\left\lfloor {\sqrt n } \right\rfloor }^3}}}} = \sum\limits_{k = 1}^{ + \infty } {\sum\limits_{n = {k^2}}^{{{\left( {k + 1} \right)}^2} - 1} {\frac{{{{\left( { - 1} \right)}^n}\left( {n - 1} \right)}}{{{{\left\lfloor {\sqrt n } \right\rfloor }^3}}}} } = \mathop = \limits^{\left[ 2 \right]} = \sum\limits_{k = 1}^{ + \infty } {\frac{1}{{{k^3}}}\sum\limits_{n = {k^2}}^{{{\left( {k + 1} \right)}^2} - 1} {{{\left( { - 1} \right)}^n}\left( {n - 1} \right)} } = }")
![\displaystyle{ = \mathop = \limits^{\left[ 3 \right]} = \sum\limits_{k = 1}^{ + \infty } {\frac{{{{\left( { - 1} \right)}^k}\left( {k\left( {k + 1} \right) - 1} \right)}}{{{k^3}}}} = \sum\limits_{k = 1}^{ + \infty } {\frac{{{{\left( { - 1} \right)}^k}\left( {{k^2} + k - 1} \right)}}{{{k^3}}}} = \sum\limits_{k = 1}^{ + \infty } {\frac{{{{\left( { - 1} \right)}^k}}}{k}} + \sum\limits_{k = 1}^{ + \infty } {\frac{{{{\left( { - 1} \right)}^k}}}{{{k^2}}}} - \sum\limits_{k = 1}^{ + \infty } {\frac{{{{\left( { - 1} \right)}^k}}}{{{k^3}}}} = \mathop = \limits^{\left[ 4 \right]} = - \ln 2 - \frac{{{\pi ^2}}}{{12}} + \frac{3}{4}\zeta \left( 3 \right)}](/forum/ext/geomar/texintegr/latexrender/pictures/54f70ff700ed2f104e8a6e2efd69d1f8.png "\displaystyle{ = \mathop = \limits^{\left[ 3 \right]} = \sum\limits_{k = 1}^{ + \infty } {\frac{{{{\left( { - 1} \right)}^k}\left( {k\left( {k + 1} \right) - 1} \right)}}{{{k^3}}}} = \sum\limits_{k = 1}^{ + \infty } {\frac{{{{\left( { - 1} \right)}^k}\left( {{k^2} + k - 1} \right)}}{{{k^3}}}} = \sum\limits_{k = 1}^{ + \infty } {\frac{{{{\left( { - 1} \right)}^k}}}{k}} + \sum\limits_{k = 1}^{ + \infty } {\frac{{{{\left( { - 1} \right)}^k}}}{{{k^2}}}} - \sum\limits_{k = 1}^{ + \infty } {\frac{{{{\left( { - 1} \right)}^k}}}{{{k^3}}}} = \mathop = \limits^{\left[ 4 \right]} = - \ln 2 - \frac{{{\pi ^2}}}{{12}} + \frac{3}{4}\zeta \left( 3 \right)}")
![\displaystyle{\left[ 1 \right]}](/forum/ext/geomar/texintegr/latexrender/pictures/13f1e05370a56e5ca6c9fc12ce7f3155.png "\displaystyle{\left[ 1 \right]}")
Για κάθε θετικό ακέραιο αριθμό 
, υπάρχουν 
τρόποι να γραφεί σαν άθροισμα δυο διακριτών θετικών ακεραίων π.χ. 
.![\displaystyle{\left[ 2 \right]}](/forum/ext/geomar/texintegr/latexrender/pictures/ab452b1852166d3e3bca72515ad82c5e.png "\displaystyle{\left[ 2 \right]}")
Για κάθε θετικό ακέραιο 
ισχύει 
.![\displaystyle{\left[ 3 \right]}](/forum/ext/geomar/texintegr/latexrender/pictures/18df5c721a4f5d3270136ee0e524a394.png "\displaystyle{\left[ 3 \right]}")
Το άθροισμα 
είναι άθροισμα όρων αριθμητικής προόδου.![\displaystyle{\left[ 4 \right]}](/forum/ext/geomar/texintegr/latexrender/pictures/e192af02525097367898a2e05b3f3a42.png "\displaystyle{\left[ 4 \right]}")
Οι υπόψη σειρές είναι γνωστές, έχουν συζητηθεί και υπολογιστεί πολλές φορές.![[1]](/forum/ext/geomar/texintegr/latexrender/pictures/35dba5d75538a9bbe0b4da4422759a0e.png "[1]")
η άθροιση κατά τρίγωνα. 
