Sigma Notation and Series Properties

When working with long sequences of numbers, writing out every term of a series can be tedious. Sigma notation provides a compact way to represent the sum of a sequence.

What is Sigma Notation?

The Greek capital letter Sigma ($\Sigma$) is used in mathematics to denote a sum. A typical series written in sigma notation looks like this: $\sum_{k=1}^{n} a_k$

• $k$ is the index of summation (often represented by $i$, $j$, or $k$).
• $k=1$ is the lower limit, telling you the starting value of the index.
• $n$ is the upper limit, telling you the final value of the index.
• $a_k$ is the general formula for the terms being added.

Example: Write $3 + 6 + 9 + \cdots + 60$ using sigma notation. Notice that each term is a multiple of 3: $3(1) + 3(2) + 3(3) + \cdots + 3(20)$. We can write this compactly as: $\sum_{k=1}^{20} 3k$

Properties of Series (Linearity)

Sigma notation follows specific algebraic rules that make it easier to manipulate and evaluate sums.

1. Constant Multiple Rule: You can factor a constant out of a summation. $\sum_{k=1}^{n} c \cdot a_k = c \sum_{k=1}^{n} a_k$

2. Addition and Subtraction Rules: The sum of a sum (or difference) is the sum (or difference) of the sums. $\sum_{k=1}^{n} (a_k \pm b_k) = \sum_{k=1}^{n} a_k \pm \sum_{k=1}^{n} b_k$

Common Summation Formulas

Instead of adding terms one by one, you can use these standard formulas for common series starting at $k=1$:

• Sum of a constant: $\sum_{k=1}^{n} c = cn$
• Sum of the first $n$ integers: $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$
• Sum of the first $n$ squares: $\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$

Example Problem

Evaluate $\sum_{k=1}^{50} k$

This notation asks for the sum of the first 50 positive integers: $1 + 2 + 3 + \cdots + 50$. Instead of adding them manually, use the formula for the sum of the first $n$ integers, where $n = 50$: $\sum_{k=1}^{50} k = \frac{50(50 + 1)}{2}$ $= \frac{50 \times 51}{2}$ $= 25 \times 51 = 1275$

The sum of the series is $1275$.