Method of Differences

A technique for evaluating finite sums where terms cancel in a telescoping pattern. Particularly useful for rational functions that can be decomposed using partial fractions.

Core Principle

If a general term $a_k$ can be expressed as:

$$a_k = f(k) - f(k-1)$$

Then the sum becomes:

$$\sum_{k=1}^{n} a_k = \sum_{k=1}^{n} [f(k) - f(k-1)]$$

When expanded, most terms cancel:

$$= [f(1) - f(0)] + [f(2) - f(1)] + [f(3) - f(2)] + \ldots + [f(n) - f(n-1)]$$

$$= f(n) - f(0)$$

This is called a telescoping series because it collapses like a telescope.

Partial Fractions Decomposition

The method of differences often requires expressing rational functions as partial fractions.

Example Decomposition

$$\frac{1}{(k+1)(k+2)} = \frac{1}{k+1} - \frac{1}{k+2}$$

Verification: $$\frac{1}{k+1} - \frac{1}{k+2} = \frac{(k+2) - (k+1)}{(k+1)(k+2)} = \frac{1}{(k+1)(k+2)} \checkmark$$

Worked Example

Evaluate $\sum_{k=1}^{n} \frac{1}{(k+1)(k+2)}$

Step 1: Decompose using partial fractions $$\frac{1}{(k+1)(k+2)} = \frac{1}{k+1} - \frac{1}{k+2}$$

Step 2: Write out the telescoping sum $$\sum_{k=1}^{n} \left(\frac{1}{k+1} - \frac{1}{k+2}\right)$$

Step 3: Expand to see the cancellation pattern

$k$ First Part Second Part
1 $\frac{1}{2}$ $-\frac{1}{3}$
2 $\frac{1}{3}$ $-\frac{1}{4}$
3 $\frac{1}{4}$ $-\frac{1}{5}$
... ... ...
$n$ $\frac{1}{n+1}$ $-\frac{1}{n+2}$

Step 4: After cancellation, only the first term of the first part and the last term of the second part remain:

$$= \frac{1}{2} - \frac{1}{n+2}$$

Step 5: Simplify $$= \frac{(n+2) - 2}{2(n+2)} = \frac{n}{2(n+2)}$$

Common Patterns

Original Term Partial Fraction Form Telescopes To
$\frac{1}{k(k+1)}$ $\frac{1}{k} - \frac{1}{k+1}$ $1 - \frac{1}{n+1}$
$\frac{1}{k(k+2)}$ $\frac{1}{2}\left(\frac{1}{k} - \frac{1}{k+2}\right)$ $\frac{1}{2}\left(1 + \frac{1}{2} - \frac{1}{n+1} - \frac{1}{n+2}\right)$
$\frac{1}{(k+1)(k+2)}$ $\frac{1}{k+1} - \frac{1}{k+2}$ $\frac{1}{2} - \frac{1}{n+2}$

When to Use Method of Differences

✅ Use when:

  • Terms can be written as $f(k) - f(k-1)$
  • Rational functions that decompose nicely via partial fractions
  • Products in the denominator with linear factors

❌ Don't use when:

  • Polynomial sums (use Summation Formulas instead)
  • Geometric series (use geometric series formula)
  • Terms don't telescope

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