Doppelsummen, Summenformeln zur schnelleren Berechnung

Question

5       10   

∑       ∑ 1/5 i^2 j.           Folgende Doppelsumme soll händisch berechnet werden. Antwort ist 605 und ich kann es auch

i=1   j=1                        berechnen, leider nur indem ich jedes i und j einzeln einsetze und dann 50 Terme habe... das müsste doch auch deutlich schneller gehen durch das Nutzen gewisser Summenformeln? Danke

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Answer 1

Wenn man sich die Summe aufschreibt, sollte man doch die Struktur erkennen. Das bekommt man auch hin, ohne die Summenformeln zu kennen, indem man Rechengesetze ausnutzt. Es gilt $$\sum_{i=1}^{5}\sum_{j=1}^{10}\frac{1}{5}i^2j =\frac{1}{5}\sum_{i=1}^{5}i^2\sum_{j=1}^{10}j=\frac{1}{5}\cdot \frac{5\cdot 6\cdot 11}{6}\cdot \frac{10\cdot 11}{2}=11\cdot 5\cdot 11 = 605.$$

Hier wurden die Summenformeln von Gauß und für Quadratzahlen genutzt. Das sind hier die Terme \(\frac{i(i+1)(2i+1)}{6}\) und \(\frac{j(j+1)}{2}\).