The method of differences

What you’ll learn

• Express summands as cancelling differences.
• Find finite and convergent infinite sums.

Understand the idea

A telescoping sum has consecutive terms that cancel after a summand is written as a difference. Partial fractions often produce this structure. Write out the first few and last few terms before cancelling so that the surviving endpoints are clear.

For a sum to infinity, first obtain the exact finite sum $S_n$ and then take its limit as $n$ tends to infinity. The uncancelled tail terms decide whether this limit exists; a pattern of cancellation alone does not prove convergence.

Key formulae

$$\frac1{r(r+1)}=\frac1r-\frac1{r+1}$$

$$\sum_{r=1}^n\bigl(f(r)-f(r+1)\bigr)=f(1)-f(n+1)$$

$$S_\infty=\lim_{n\to\infty}S_n$$

Worked example

Find $\sum_{r=1}^n\frac2{(r+1)(r+2)}$ and its limit.

1. The summand is $2/(r+1)-2/(r+2)$.
2. Expanding gives $(1-2/3)+(2/3-1/2)+\cdots+(2/(n+1)-2/(n+2))$.
3. Everything cancels except $1-2/(n+2)$.
4. The tail tends to zero as $n$ tends to infinity.

Answer: $S_n=1-\frac2{n+2}$ and $S_\infty=1$.