Summation Notation Overview

Overview

This lecture explains summation notation (Sigma notation), its components, and provides several worked-out examples for evaluating different sums.

Sigma (Σ) notation is a concise way to represent the sum of a sequence of terms.
The symbol Σ indicates summation, or adding together all specified terms.
Sigma notation has three parts: the lower limit (starting value), upper limit (ending value), and the expression to sum.

The lower limit (e.g., k = 1) is where the index of summation begins.
The upper limit (e.g., n = 4) is where the index ends.
The expression (e.g., 5k) is the formula to substitute and sum for each index value.

For Σ (from k=1 to 4) of 5k: Substitute k = 1, 2, 3, 4 and sum 5×1, 5×2, 5×3, 5×4; sum is 50.
For Σ (from k=1 to 6) of (3k + 1): Substitute k = 1 to 6, calculate each, then add; sum is 69.
For Σ (from k=0 to 4) of k²: Substitute k = 0 to 4, square each, and sum; total is 30.
For Σ (from k=1 to 5) of (–1)^(k+1): Substitute and check if exponent is even (result is 1) or odd (result is –1); sum is 1.
For Σ (from k=0 to 3) of k³/(k+1): Substitute k = 0 to 3 into numerator and denominator, sum the resulting fractions.
For Σ (from k=1 to 5) of (–1)^k/k: Substitute k = 1 to 5 and alternate sign depending on k being odd or even; final simplified sum is –47/60.

If the exponent of (–1) is even, result is positive; if odd, result is negative.
For fractional sums, substitute each value and simplify, sometimes finding a common denominator (LCD) when necessary.

Sigma Notation (Σ) — a mathematical symbol used to indicate summing a series of terms.
Index of Summation — the variable (often k) that takes values from the lower to the upper limit in a summation.
Lower Limit — the starting value of the index.
Upper Limit — the ending value of the index.
Summand — the expression to substitute and sum.

Practice evaluating sums using Sigma notation with different expressions.
Review and memorize the rules for evaluating (–1)^k and summing series involving fractions.