11.4Series and Their Notations
Use sigma notation Σ to represent sums, evaluate arithmetic and geometric series, and determine when an infinite geometric series converges.
Series are the bridge between sequences and calculus. Infinite series underpin Taylor series, Fourier analysis, and numerical methods — understanding convergence is essential for any STEM field.
Essential Question
How can sigma notation compactly represent the sum of many terms, and what formulas let us find those sums efficiently?
Overview
A series is the sum of the terms of a sequence. When a sequence has many terms, writing out every addition is tedious. Sigma (Σ) notation gives us a compact way to express any sum. The Greek capital letter Σ means "sum of," and the expression below and above it tells us where to start and stop.
The partial sum Sn is the sum of the first n terms. For arithmetic and geometric sequences, closed-form formulas let us compute Sn without listing every term. For infinite geometric series with |r| < 1, the sum converges to a finite value S∞ = a₁ / (1 − r).
Anatomy of sigma notation: Σ(k=1 to n) ak
Key Formulas
- Arithmetic series: Sn = n/2 · (a₁ + an) = n/2 · (2a₁ + (n−1)d)
- Finite geometric series: Sn = a₁(1 − rⁿ) / (1 − r), r ≠ 1
- Infinite geometric series: S∞ = a₁ / (1 − r) when |r| < 1; diverges when |r| ≥ 1
Worked Examples
Evaluate Σ(k=1 to 5) (2k − 1) by listing terms and summing.
Substitute k = 1: 2(1) − 1 = 1
Substitute k = 2: 2(2) − 1 = 3
Substitute k = 3: 2(3) − 1 = 5
Substitute k = 4: 2(4) − 1 = 7
Substitute k = 5: 2(5) − 1 = 9
Sum all terms: 1 + 3 + 5 + 7 + 9 = 25
Find S₁₀ for the arithmetic series with a₁ = 3 and d = 4.
Identify: a₁ = 3, d = 4, n = 10
Find a₁₀: a₁₀ = 3 + (10 − 1)(4) = 3 + 36 = 39
Apply formula: S₁₀ = 10/2 · (a₁ + a₁₀) = 5 · (3 + 39)
S₁₀ = 5 · 42 = 210
Find S₆ for the geometric series with a₁ = 2 and r = 3.
Identify: a₁ = 2, r = 3, n = 6
Apply formula: S₆ = a₁(1 − r⁶) / (1 − r)
Compute r⁶ = 3⁶ = 729
S₆ = 2(1 − 729) / (1 − 3) = 2(−728) / (−2)
S₆ = −1456 / −2 = 728
Find S∞ for the infinite geometric series with a₁ = 8 and r = 1/2.
Check convergence: |r| = |1/2| = 0.5 < 1 ✓ — series converges
Apply formula: S∞ = a₁ / (1 − r)
S∞ = 8 / (1 − 1/2) = 8 / (1/2)
S∞ = 8 × 2 = 16
Write sigma notation for the series 4 + 8 + 12 + 16 + 20 + 24.
Identify the pattern: each term is a multiple of 4
Term k: aₖ = 4k
First term: k = 1 gives 4(1) = 4 ✓
Last term: k = 6 gives 4(6) = 24 ✓
Write in sigma notation: Σ(k=1 to 6) 4k
Guided Practice
Evaluate Σ(k=1 to 4) (3k + 1) by listing each term then summing.
Hint: Substitute k = 1, 2, 3, 4 into (3k + 1) to get four terms, then add them.
Find S₈ for the arithmetic series with a₁ = 5 and d = 3.
Hint: First find a₈ using aₙ = a₁ + (n−1)d, then use S₈ = 8/2 · (a₁ + a₈).
Find S₅ for the geometric series with a₁ = 1 and r = 4.
Hint: Use Sₙ = a₁(1 − rⁿ)/(1 − r). Compute 4⁵ = 1024 first.
Find S∞ for the infinite geometric series with a₁ = 6 and r = 1/3.
Hint: Check |r| < 1, then apply S∞ = a₁/(1 − r). Simplify the fraction carefully.
Write sigma notation for 1 + 4 + 9 + 16 + 25.
Hint: Notice each term is a perfect square: 1², 2², 3², 4², 5². What is the general term aₖ?
Common Mistakes
Using the infinite series formula S∞ = a₁/(1−r) when |r| ≥ 1.
Always check |r| < 1 first. If |r| ≥ 1, the series diverges and has no finite sum.
Forgetting to include the lower bound when evaluating sigma notation (starting at k = 0 instead of k = 1).
Always read the lower bound carefully — it tells you the first value of k to substitute.
Using the arithmetic formula Sₙ = n/2·(a₁+aₙ) for a geometric series.
Match the formula to the series type: arithmetic uses d, geometric uses r.
Computing rⁿ as r·n (multiplying instead of raising to a power).
rⁿ means r multiplied by itself n times: e.g., 3⁴ = 81, not 3×4 = 12.
Math Tips
For small sigma sums (n ≤ 6), listing every term and adding is often faster and less error-prone than using a formula.
The arithmetic series formula Sₙ = n/2·(a₁+aₙ) is easy to remember: it's the average of the first and last terms, times the number of terms.
For infinite geometric series, think of S∞ = a₁/(1−r) as 'first term divided by (1 minus ratio).'
When writing sigma notation, always verify your formula by plugging in the first and last index values.
A geometric series with r = −1/2 still converges because |−1/2| = 0.5 < 1 — the sign of r doesn't matter for convergence, only the absolute value.
Series
The sum of the terms of a sequence.
Sigma notation (Σ)
A compact notation using the Greek letter Σ to represent a sum, with an index variable, lower bound, and upper bound.
Index variable
The variable (often k or i) that takes on each integer value from the lower bound to the upper bound.
Partial sum (Sₙ)
The sum of the first n terms of a series.
Arithmetic series
The sum of an arithmetic sequence; computed with Sₙ = n/2·(a₁+aₙ).
Geometric series
The sum of a geometric sequence; computed with Sₙ = a₁(1−rⁿ)/(1−r).
Infinite geometric series
A geometric series with infinitely many terms; converges to S∞ = a₁/(1−r) when |r| < 1.
Convergent series
An infinite series whose partial sums approach a finite limit.
Divergent series
An infinite series whose partial sums grow without bound (no finite sum).
Interactive Practice — 1 Questions
Which formula gives the sum of an infinite geometric series when |r| < 1?