Series, Sigma Notation & Convergence (Grade 12)

Sum formulae for arithmetic and geometric series, reading and writing sigma notation, and the sum to infinity — including the values of the ratio for which a series converges.

A sequence is a list of terms; a series is what you get when you add them. Sₙ means the sum of the first n terms.

1The formulae

Arithmetic
Tₙ = a + (n − 1)d · Sₙ = n2[2a + (n − 1)d]

Geometric
Tₙ = arⁿ⁻¹ · Sₙ = a(rⁿ − 1)r − 1, r ≠ 1

💡

If you already know the last term ℓ, the arithmetic sum is quicker as Sₙ = n2(a + ℓ).

Worked ExampleTₙ = 2n + 4. Determine S₁₉

1
a = T₁ = 2(1) + 4 = 6, and d = 2.
The coefficient of n is the common difference.
2
S₁₉ = 192[2(6) + 18(2)].
Substitute into the arithmetic sum formula.
3
= 192[12 + 36] = 192(48).
4
= 456.

2Sigma notation

∑_n=1^k Tₙ = T₁ + T₂ + … + Tₖ

The letter below the ∑ is the counter and its starting value; the number on top is where it stops. Number of terms = top − bottom + 1.

Worked ExampleEvaluate ∑_n=1²⁰ (3n − 1)

1
This is arithmetic: a = 3(1) − 1 = 2, d = 3, n = 20.
20 − 1 + 1 = 20 terms.
2
S₂₀ = 202[2(2) + 19(3)] = 10[4 + 57].
3
= 10(61) = 610.

⚠️

∑_n=3¹⁰ has 8 terms, not 7 — it is 10 − 3 + 1. Getting the number of terms wrong loses every mark that follows.

3Writing a series in sigma notation

Worked ExampleWrite 2 + 5 + 8 + … + 59 in sigma notation

1
Arithmetic with a = 2, d = 3, so Tₙ = 3n − 1.
a + (n−1)d = 2 + 3n − 3.
2
Find where it stops: 3n − 1 = 59 → n = 20.
3
∑_n=1²⁰ (3n − 1).

4Sum to infinity and convergence

S_∞ = a1 − r only when −1 < r < 1

If |r| < 1 each term is smaller than the last, so the total settles on a fixed value — the series converges. Otherwise the terms grow and it diverges.

Worked ExampleDetermine S_∞ for 8 + 4 + 2 + 1 + …

1
r = 48 = 12, and |r| < 1, so it converges.
Always state this check.
2
S_∞ = 81 − 12 = 812.
3
= 16.

Worked ExampleFor which values of x does ∑ 3(2x)ⁿ⁻¹ converge?

1
Here r = 2x.
The base of the power is the ratio.
2
Need −1 < 2x < 1.
The convergence condition.
3
−12 < x < 12.
Divide throughout by 2.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
The general term of an arithmetic sequence is Tₙ = 2n + 4. Determine T₁₀.
Show answer ▾Hide answer ▴
2(10) + 4 = 24
2
Determine the sum of the first 19 terms of Tₙ = 2n + 4.
Show answer ▾Hide answer ▴
a = 6, d = 2. S₁₉ = 192[12 + 36] = 456
3
Evaluate ∑_n=1²⁰ (3n − 1)
Show answer ▾Hide answer ▴
a = 2, d = 3, n = 20. S = 10[4 + 57] = 610
4
How many terms are there in ∑_n=3¹⁰ Tₙ?
Show answer ▾Hide answer ▴
10 − 3 + 1 = 8 terms
5
Write 2 + 5 + 8 + … + 59 in sigma notation.
Show answer ▾Hide answer ▴
Tₙ = 3n − 1 and 3n − 1 = 59 gives n = 20, so ∑_n=1²⁰ (3n − 1)
6
Determine S_∞ for 8 + 4 + 2 + 1 + …
Show answer ▾Hide answer ▴
r = 12, so S_∞ = 81 − 12 = 16
7
Determine the values of x for which ∑ 3(2x)ⁿ⁻¹ converges.
Show answer ▾Hide answer ▴
−1 < 2x < 1, so −12 < x < 12
8
Calculate S₁₀ for the geometric series 2 + 6 + 18 + …
Show answer ▾Hide answer ▴
a = 2, r = 3. S₁₀ = 2(3¹⁰ − 1)2 = 59 048