Summation (Sigma) Notation Overview

Overview

This lecture explains summation (Sigma) notation, its parts, and demonstrates how to evaluate sums using several example problems.

Sigma (Σ) is a Greek letter used to represent summation (the sum of terms in a sequence).
Summation notation simplifies writing the addition of many terms, such as a₁ + a₂ + ... + aₙ.
The symbol includes the index of summation, lower limit (start), upper limit (end), and the expression to sum.

The lower limit indicates where to start substituting the index value.
The upper limit shows the last value of the index to substitute.
The index of summation (e.g., "k") is the variable that changes from lower to upper limit.
The expression tells what to calculate and add as the index varies.

Example 1: Σ from k=1 to 4 of 5k = 5×1 + 5×2 + 5×3 + 5×4 = 50.
Example 2: Σ from k=1 to 6 of (3k+1) = 4 + 7 + 10 + 13 + 16 + 19 = 69.
Example 3: Σ from k=0 to 4 of k² = 0² + 1² + 2² + 3² + 4² = 30.
Example 4: Σ from k=1 to 5 of (–1)^(k+1)
- Substitute k: exponents are 2, 3, 4, 5, 6 → results: 1, –1, 1, –1, 1
- Summing gives 1.
Example 5: Σ from k=0 to 3 of k³/(k+1)
- Substitute k and calculate each term: 0/1, 1/2, 8/3, 27/4. Find the sum using a common denominator.
Example 6: Σ from k=1 to 5 of (–1)^k / k
- Results alternate between negative and positive fractions: –1, 1/2, –1/3, 1/4, –1/5
- Sum using a common denominator: –47/60.

An even exponent of (–1) gives a positive result; an odd exponent gives a negative result.
Always substitute each index value and sum the results.

Summation (Σ) notation — A mathematical shorthand for summing a sequence of terms.
Index of summation — The variable that changes value in the sum (usually k, i, or n).
Lower limit — The starting value of the index in the summation.
Upper limit — The ending value for the index in the summation.
Expression — The formula or value to be evaluated for each index.