Q. 3 - Calculus

Question

Explain why $\sum _{i=1}^l \sum _{j=1}^m \sum _{k=1}^n i{j}^2{k}^3=\sum _{j=1}^m \sum _{k=1}^n \sum _{i=1}^l i{j}^2{k}^3$.

Step-by-Step Solution

Answer

The two summations $\sum _{i=1}^l \sum _{j=1}^m \sum _{k=1}^n i{j}^2{k}^3=\sum _{j=1}^m \sum _{k=1}^n \sum _{i=1}^l i{j}^2{k}^3$ contains the same summands although the summands are rearranged by the cumulative and associative properties of addition, they are equal.

Step 1 . Given information

We need to explain why $\sum _{i=1}^l \sum _{j=1}^m \sum _{k=1}^n i{j}^2{k}^3=\sum _{j=1}^m \sum _{k=1}^n \sum _{i=1}^l i{j}^2{k}^3$.

Step 2 . Definition for the summation:

The summation definition states that if the order of the summation will change but there is no effect on the result. The two summations $\sum _{i=1}^l \sum _{j=1}^m \sum _{k=1}^n i{j}^2{k}^3=\sum _{j=1}^m \sum _{k=1}^n \sum _{i=1}^l i{j}^2{k}^3$contains the same summands although the summands are rearranged by the cumulative and associative properties of addition, they are equal.