What Is a Series?

A series is the sum of the terms of a sequence. While a sequence lists terms, a series adds them together. The sum of the first n terms is called the nth partial sum, denoted S_n.

Sequence: 2, 5, 8, 11, 14

Series: S₅ = 2 + 5 + 8 + 11 + 14 = 40

Arithmetic Series Formula

For an arithmetic series with first term a, common difference d, and n terms:

Using first and last terms:

S_n = n/2 (a + l)

where l = last term

Using a and d:

S_n = n/2 (2a + (n-1)d)

when last term is unknown

Geometric Series and Sigma Notation

For a geometric series with first term a, common ratio r (where r is not equal to 1), and n terms:

S_n = a(rⁿ - 1) / (r - 1)

or equivalently S_n = a(1 - rⁿ) / (1 - r) when |r| < 1

Sigma notation provides a compact way to write a series. The Greek letter Σ (sigma) means "sum":

Σ_k=1ⁿ T_k = T₁ + T₂ + T₃ + ... + T_n

Σ_k=1⁵ (2k + 1) = 3 + 5 + 7 + 9 + 11 = 35

Σ_k=0⁴ 3^k = 1 + 3 + 9 + 27 + 81 = 121

Key Vocabulary

Series

The sum of terms in a sequence. An arithmetic series adds AP terms; a geometric series adds GP terms.

Partial Sum (S_n)

The sum of the first n terms of a series: S_n = T₁ + T₂ + ... + T_n.

Sigma Notation (Σ)

A compact notation for writing sums using the Greek letter sigma, with an index variable, lower bound, and upper bound.

Index Variable

The variable (often k, i, or n) that counts through the terms in sigma notation, running from the lower bound to the upper bound.

Worked Examples

1

Find the sum of the first 20 terms of the AP: 3, 7, 11, 15, ...

Step 1: a = 3, d = 4, n = 20.

Step 2: S₂₀ = 20/2 (2(3) + (20-1)(4)) = 10(6 + 76) = 10(82) = 820.

Answer: S₂₀ = 820.

2

Find the sum of the first 6 terms of the GP: 2, 6, 18, 54, ...

Step 1: a = 2, r = 3, n = 6.

Step 2: S₆ = 2(3⁶ - 1)/(3 - 1) = 2(729 - 1)/2 = 728.

Answer: S₆ = 728.

3

Evaluate Σ_k=1⁴ (3k - 1).

Step 1: Substitute k = 1, 2, 3, 4 into (3k - 1):

k=1: 3(1)-1 = 2; k=2: 3(2)-1 = 5; k=3: 3(3)-1 = 8; k=4: 3(4)-1 = 11.

Step 2: Sum = 2 + 5 + 8 + 11 = 26.

Answer: Σ_k=1⁴ (3k - 1) = 26.

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

Find S₁₀ for the AP: 1, 4, 7, 10, ...

145 155 130 28

Question 2

What does Σ_k=1³ k² equal?

6 14 9 36

Question 3

Find S₅ for the GP: 4, 12, 36, 108, ...

364 324 484 484

Question 4

Express the sum 5 + 10 + 15 + 20 + 25 in sigma notation.

Σ_k=1²⁵ k Σ_k=1⁵ (k + 5) Σ_k=0⁵ 5k Σ_k=1⁵ 5k

Question 5

Find the sum of the first 100 positive integers (1 + 2 + 3 + ... + 100).

5050 10000 5000 4950

Series and Sigma Notation