Sum of Reciprocals of Consecutive Products

Sum of Reciprocals of Consecutive Products

## Formula

$$\sum_{k=1}^{n}\frac1{k(k+1)}=\frac n{n+1}$$

More generally:

$$\sum_{k=1}^{n}\frac1{k(k+r)}=\frac1r\left(\sum_{k=1}^{r}\frac1k-\sum_{k=n+1}^{n+r}\frac1k\right)$$

for positive integer $$r$$.

## Usage

- Used in telescoping-series problems involving rational terms.