Sigma Notation

Sigma Notation: Compact Summation

Sigma notation is a compact way to express the sum of many terms that follow a pattern. Instead of writing $f(x_1) + f(x_2) + \cdots + f(x_n)$ repeatedly, we use:

$\sum_{i=m}^{n} a_i = a_m + a_{m+1} + a_{m+2} + \cdots + a_n$

Here, $i$ is the index of summation, $m$ is the lower limit, $n$ is the upper limit, and $a_i$ is the formula determining each term.

Evaluating a Sum

To evaluate a sum, substitute each integer from the lower to upper limit into the formula and add:

$\sum_{i=1}^{4} i^2 = 1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30$

The index disappears after evaluation—the result is a single number.

Key Properties: Linearity

Sigma notation follows linearity rules that simplify complex sums:

$\sum_{i=m}^{n} c \cdot a_i = c \sum_{i=m}^{n} a_i \quad \text{(constants factor out)}$

$\sum_{i=m}^{n} (a_i + b_i) = \sum_{i=m}^{n} a_i + \sum_{i=m}^{n} b_i \quad \text{(sums split apart)}$

Closed-Form Formulas

Rather than computing term-by-term, use these standard formulas:

These enable symbolic evaluation with variable $n$.

Example

Evaluate $\sum_{i=1}^{n} (2i+3)$ using linearity:

$\sum_{i=1}^{n} (2i+3) = 2\sum_{i=1}^{n} i + \sum_{i=1}^{n} 3 = 2 \cdot \frac{n(n+1)}{2} + 3n = n^2 + 4n$

Check: For $n=2$, direct gives $(2 \cdot 1+3)+(2 \cdot 2+3) = 12$; formula gives $4+8=12$ ✓

Pitfall to Avoid

The index $i$ is a dummy variable—it exists only within the summation. Once you evaluate, the index vanishes. Confusing this can lead to errors when translating between expanded and compact forms.