Sum of Series

The sigma symbol Σ is shorthand for adding many terms together, one by one. When you add 1 to n, sums like the naturals or the squares appear so often that a short formula gives the answer at once. Once you see where these formulas come from, any messy series breaks into sums of k, k², and k³. Here a bar chart and a triangle of dots let you watch the sums build and see why the formula holds.

Meaning of the Sigma (Σ) Notation

Number of terms n5

💡 Σ means 'add them all up'

①Σ(k=1 to n) a_k = a₁ + a₂ + ... + a_n

②Bottom: start, top: end, right: rule

③Sum of bar heights = the value of Σ

Sum of Naturals — Triangular Numbers

Sum of Naturals

n(n+1)2

1 + 2 + 3 + ... + n = n(n+1)/2

Three Key Formulas

Σk

n(n+1)2

1 + 2 + ... + n

Σk²

n(n+1)(2n+1)6

1² + 2² + ... + n²

Σk³

[n(n+1)2]² = (Σk)²

1³ + 2³ + ... + n³ = (1+2+...+n)²

🔗 Surprising Identity

①Σk³ = (Σk)² — sum of cubes is the square of the sum!

②These three handle most sum problems

③Decompose any expression into k, k², k³

Properties of Σ

Linearity

Σ(ca_k + b_k) = c·Σa_k + Σb_k

Constants pull out, sums split

Recovering a_n

a_n = S_n - S_n-1 (n ≥ 2)

Get the general term back from the partial sum

⚠️ Watch Out

①n ≥ 2: a_n = S_n − S_{n-1}

②n = 1: a₁ = S₁ (check separately!)

③Verify the two agree

Wrap-up

Three Sum Formulas

Σk = n(n+1)2, Σk² = n(n+1)(2n+1)6, Σk³ = [n(n+1)2]²

Sums of naturals, squares, cubes

🎯 Exam Points

①Σk = n(n+1)/2 — basic

②Σk² = n(n+1)(2n+1)/6

③Σk³ = [n(n+1)/2]² = (Σk)²

④Σ is linear: pull out constants, split sums

⑤a_n = S_n − S_{n-1} (n ≥ 2); check a₁ = S₁ separately

Worked Examples & Past Exam

Example 1

Find the value of 1 + 2 + 3 + ... + 10.

Substitute n = 10 into the sum formula Σk = n(n+1)/2.

Σk = 10 × 112

Compute.

= 1102 = 55

▸ 55

The sum of the first n natural numbers is n(n+1)/2.

Example 2

Find the value of 1² + 2² + 3² + 4².

Substitute n = 4 into the sum formula Σk² = n(n+1)(2n+1)/6.

Σk² = 4 × 5 × 96

Compute.

= 1806 = 30

▸ 30

The sum of squares is n(n+1)(2n+1)/6.

2023 CSAT Math type, adapted

Express Σ(k=1 to n) (2k + 1) as an expression in n.

①n(n+2)

②n(n+1)

③n²

④2n+1

⑤n²+1

▸ ① n(n+2)

Split using linearity: Σ(2k+1) = 2Σk + Σ1.

Σ(2k+1) = 2 × n(n+1)2 + n

Simplify.

= n(n+1) + n = n² + 2n = n(n+2)

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