Summation Shifting

Shifting the Index

• This is often done when two sums need to be added but their indices of summation do not match.
• When shifting an index of summation, make the appropriate changes in the corresponding summand.

Example

Suppose the summation is ${\sum }_{j=1}^5{j}^2$ but we want the index to be between 0 to 4.

Let $k=j-1$, and the term $j^2$ becomes $(k+1{)}^2$.

${\sum }_{j=1}^5{j}^2={\sum }_{k=0}^4(k+1{)}^2$