Method of Differences

The method of differences (also called telescoping) is a technique for summing series where most terms cancel. It is used when the general term can be written as a difference $f(r) - f(r+1)$f(r)−f(r+1) (or similar), causing consecutive terms to cancel in the sum.

---

The Key Idea

If we can write the general term as:

$u_r = f(r) - f(r+1)$ur​=f(r)−f(r+1)

then:

$\sum_{r=1}^{n} u_r = [f(1) - f(2)] + [f(2) - f(3)] + \cdots + [f(n) - f(n+1)] = f(1) - f(n+1)$∑r=1n​ur​=[f(1)−f(2)]+[f(2)−f(3)]+⋯+[f(n)−f(n+1)]=f(1)−f(n+1)

Most intermediate terms cancel — this is called telescoping.

---

Using Partial Fractions

The most common application is to decompose a fraction using partial fractions, then telescope.

Worked Example 1: Find $\sum_{r=1}^{n} \frac{1}{r(r+1)}$∑r=1n​r(r+1)1​.

Partial fractions: $\frac{1}{r(r+1)} = \frac{1}{r} - \frac{1}{r+1}$r(r+1)1​=r1​−r+11​.

$\sum_{r=1}^{n} \left(\frac{1}{r} - \frac{1}{r+1}\right) = \left(1 - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \cdots + \left(\frac{1}{n} - \frac{1}{n+1}\right)$∑r=1n​(r1​−r+11​)=(1−21​)+(21​−31​)+⋯+(n1​−n+11​)

$= 1 - \frac{1}{n+1} = \frac{n}{n+1}$=1−n+11​=n+1n​

Worked Example 2: Find $\sum_{r=1}^{n} \frac{1}{r(r+2)}$∑r=1n​r(r+2)1​.

$\frac{1}{r(r+2)} = \frac{1}{2}\left(\frac{1}{r} - \frac{1}{r+2}\right)$r(r+2)1​=21​(r1​−r+21​) (cover-up rule).

$\sum_{r=1}^{n} \frac{1}{2}\left(\frac{1}{r} - \frac{1}{r+2}\right)$∑r=1n​21​(r1​−r+21​)

Write out the first few and last few terms: