Algebra • Topic 10

Sigma Notation

Sigma notation ( $\sum$ ) is the standard way to write long sums compactly. Mastery of its properties is essential for series, combinatorics, and number theory.

Definition

Definition (Sigma Notation)

i = m \sum n a_{i} = a_{m} + a_{m + 1} + \dots + a_{n}

$i$ is the index of summation.
$m$ is the lower bound.
$n$ is the upper bound.

Properties

1. Linearity

i = 1 \sum n (c a_{i} + b_{i}) = c i = 1 \sum n a_{i} + i = 1 \sum n b_{i}

(Constants can be pulled out; sums can be split).

2. Telescoping Sums

This is a critical Olympiad technique for evaluating sums. If

a_{i}

can be written as

b_{i + 1} - b_{i}

(or

b_{i} - b_{i + 1}

), most terms cancel out.

Example (Telescoping Series)

Evaluate

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

Proof. Use partial fraction decomposition: $\frac{1}{k ( k + 1 )} = \frac{1}{k} - \frac{1}{k + 1}$ .

k = 1 \sum n (\frac{1}{k} - \frac{1}{k + 1}) = (1 - \frac{1}{2}) + (\frac{1}{2} - \frac{1}{3}) + \dots + (\frac{1}{n} - \frac{1}{n + 1}) = 1 - \frac{1}{n + 1} = \frac{n}{n + 1}

∎

3. Double Summation

When swapping the order of summation, be careful with the bounds.

i = 1 \sum n j = 1 \sum m a_{ij} = j = 1 \sum m i = 1 \sum n a_{ij}

If bounds depend on each other (e.g., $1 \leq i \leq j \leq n$ ), visualize the region of summation (a triangle) to swap correctly.

Common Sums

$\sum_{i = 1}^{n} c = c n$
$\sum_{i = 1}^{n} i = \frac{n ( n + 1 )}{2}$
$\sum_{i = 1}^{n} i^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}$
$\sum_{i = 1}^{n} i^{3} = (\frac{n ( n + 1 )}{2})^{2}$

Practice Problems

Exercise (Problem 1)

Evaluate

\sum_{k = 1}^{20} (2 k - 1)

Exercise (Problem 2)

Simplify

\sum_{k = 1}^{n} \frac{1}{k + k + 1}

by rationalizing the denominator.

Exercise (Problem 3)

Compute

\sum_{i = 1}^{n} \sum_{j = 1}^{n} (i + j)