Question Number 194610 by justenspi last updated on 11/Jul/23
$${where}\:{can}\:{I}\:{learn}\:{about}\:{multiple}\:{sigma}\:{notaions} \\ $$$${of}\:{dependent}\:{and}\:{independent}\:{variables} \\ $$$$ \\ $$$${something}\:{like}\:{this} \\ $$$$\underset{\mathrm{1}\leqslant{i}} {\sum}\underset{<{j}} {\sum}\underset{<{k}\leqslant\mathrm{1}} {\sum}\left({i}+{j}+{k}\right)=\lambda \\ $$$${find}\:\lambda \\ $$$${I}\:{want}\:{to}\:{know}\:{what}\:{to}\:{study} \\ $$
Answered by qaz last updated on 11/Jul/23
 =[1≤k≤n][i+1≤j≤k−1][1≤i≤j−1](i+j+k) =[3≤k≤n][2≤j≤k−1][1≤i≤j−1](i+j+k) =[3≤k≤n][2≤j≤k−1]((1/2)j(j−1)+(j−1)(j+k)) =[3≤k≤n](k^3 −3k^2 +2k) =(1/4)(n−2)(n−1)n(n+1) ,(n∈N , n≥3)](https://www.tinkutara.com/question/Q194621.png)
$$\underset{\mathrm{1}\leqslant{i}<{j}<{k}\leqslant{n}} {\sum}\left({i}+{j}+{k}\right)=\left[\mathrm{1}\leqslant{i}<{j}<{k}\leqslant{n}\right]\left({i}+{j}+{k}\right) \\ $$$$=\left[\mathrm{1}\leqslant{k}\leqslant{n}\right]\left[{i}+\mathrm{1}\leqslant{j}\leqslant{k}−\mathrm{1}\right]\left[\mathrm{1}\leqslant{i}\leqslant{j}−\mathrm{1}\right]\left({i}+{j}+{k}\right) \\ $$$$=\left[\mathrm{3}\leqslant{k}\leqslant{n}\right]\left[\mathrm{2}\leqslant{j}\leqslant{k}−\mathrm{1}\right]\left[\mathrm{1}\leqslant{i}\leqslant{j}−\mathrm{1}\right]\left({i}+{j}+{k}\right) \\ $$$$=\left[\mathrm{3}\leqslant{k}\leqslant{n}\right]\left[\mathrm{2}\leqslant{j}\leqslant{k}−\mathrm{1}\right]\left(\frac{\mathrm{1}}{\mathrm{2}}{j}\left({j}−\mathrm{1}\right)+\left({j}−\mathrm{1}\right)\left({j}+{k}\right)\right) \\ $$$$=\left[\mathrm{3}\leqslant{k}\leqslant{n}\right]\left({k}^{\mathrm{3}} −\mathrm{3}{k}^{\mathrm{2}} +\mathrm{2}{k}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\left({n}−\mathrm{2}\right)\left({n}−\mathrm{1}\right){n}\left({n}+\mathrm{1}\right)\:\:\:\:\:\:\:,\left({n}\in\mathbb{N}\:,\:{n}\geqslant\mathrm{3}\right) \\ $$
Commented by justenspi last updated on 11/Jul/23
$${thanks}\:,\:{sir} \\ $$$${where}\:{can}\:{I}\:{learn}\:{that} \\ $$$${in}\:{what}\:{book} \\ $$
Commented by qaz last updated on 11/Jul/23
$${Have}\:{you}\:{studied}\:{double}\:{or}\:{triple}\:{integrals}? \\ $$$${This}\:{is}\:{similar}\:{to}\:{exchange}\:{the}\:{order}\:{of}\:{integrals}, \\ $$$${no}\:{special}\:{books}\:{are}\:{required}… \\ $$
Commented by justenspi last updated on 11/Jul/23
$${actualy}\:{I}\:{am}\:{a}\:{high}\:{school}\:{stuudent} \\ $$