Suppose \(\sum_{r=a}^{N}f(r)=g(N)\). Replacing \(r\) by \(r+k\) gives the complete limit change
\[\sum_{r=a}^{N}f(r)=g(N)\xrightarrow{\text{substitute}\hspace{0.5em}r\text{ as}\hspace{0.5em}r+k}\sum_{r+k=a}^{r+k=N}f(r+k)=g(N)\xrightarrow{\ r=a-k,\ r=N-k\ }\sum_{r=a-k}^{N-k}f(r+k)=g(N).\]
Suggested steps: Replace \(r\) by \(r+k\). Change both limits while leaving \(g(N)\) unchanged. The upper limit follows directly by substitution; changing the lower limit may require addition or subtraction of terms.
Examples of Changing the Lower Limit
| Changing the lower limit | Changing the lower limit |
|---|---|
| \[\sum_{r=2}^{N}u_r=\left(\sum_{r=1}^{N}u_r\right)-u_1\] | \[\sum_{r=3}^{N}u_r=u_3+\sum_{r=4}^{N}u_r\] |
| \[\begin{aligned}\sum_{r=3}^{N}u_r&=u_3+u_4+\sum_{r=5}^{N}u_r\\&=\sum_{r=3}^{4}u_r+\sum_{r=5}^{N}u_r\end{aligned}\] | \[\begin{aligned}\sum_{r=5}^{N}u_r&=u_5+u_6+u_7+\sum_{r=8}^{N}u_r\\&=\sum_{r=5}^{7}u_r+\sum_{r=8}^{N}u_r\end{aligned}\] |
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