Sigma Notation: Writing Sums Compactly

Sigma notation says: add these things up.

Instead of writing 1 + 4 + 9 + 16 + 25, you write:

∑ₖ₌₁⁵ k²

The sigma (∑) means "sum." The bottom tells you where to start (k = 1). The top tells you where to stop (k = 5). The expression tells you what to add (k²).

That's sigma notation. It's not a new kind of math—it's shorthand for sums that would be tedious to write out.

Reading Sigma Notation

Every sigma expression has four parts:

1. The summation symbol: ∑ (capital Greek sigma)
2. The index and starting value: k = 1 (start at 1)
3. The ending value: 5 (stop at 5)
4. The summand: k² (what you're adding)

∑ₖ₌₁⁵ k² means: "Add k² for k = 1, 2, 3, 4, 5."

k = 1: 1² = 1 k = 2: 2² = 4 k = 3: 3² = 9 k = 4: 4² = 16 k = 5: 5² = 25

Sum: 1 + 4 + 9 + 16 + 25 = 55

The index variable (often k, i, j, or n) is a dummy variable—it only exists inside the sum. You could use any letter and get the same result.

Basic Examples

Sum of integers from 1 to n:

∑ₖ₌₁ⁿ k = 1 + 2 + 3 + ... + n

Sum of squares:

∑ₖ₌₁ⁿ k² = 1 + 4 + 9 + ... + n²

Sum of first n odd numbers:

∑ₖ₌₁ⁿ (2k-1) = 1 + 3 + 5 + ... + (2n-1)

Constant sum:

∑ₖ₌₁ⁿ 5 = 5 + 5 + 5 + ... (n times) = 5n

When the summand doesn't depend on k, you're just adding the same thing n times.

Properties of Sigma Notation

Constant factor pulls out:

∑ₖ₌₁ⁿ c·aₖ = c · ∑ₖ₌₁ⁿ aₖ

You can pull constant multipliers outside the sum.

Sum splits over addition:

∑ₖ₌₁ⁿ (aₖ + bₖ) = ∑ₖ₌₁ⁿ aₖ + ∑ₖ₌₁ⁿ bₖ

A sum of sums equals the sum of the sums. (Sounds obvious, but it's powerful.)

Splitting ranges:

∑ₖ₌₁ⁿ aₖ = ∑ₖ₌₁ᵐ aₖ + ∑ₖ₌ₘ₊₁ⁿ aₖ

You can break a sum into pieces at any intermediate point.

These properties let you manipulate sums algebraically—factor, distribute, recombine.

Common Sums You Should Know

Several sums appear so often they have standard formulas:

Sum of first n integers: ∑ₖ₌₁ⁿ k = n(n+1)/2

The story: young Gauss added 1 + 2 + ... + 100 by pairing 1 with 100, 2 with 99, etc. Each pair sums to 101, and there are 50 pairs. Answer: 5050.

Sum of first n squares: ∑ₖ₌₁ⁿ k² = n(n+1)(2n+1)/6

Sum of first n cubes: ∑ₖ₌₁ⁿ k³ = [n(n+1)/2]² = (∑ₖ₌₁ⁿ k)²

The sum of cubes equals the square of the sum of integers. Beautiful.

Geometric sum: ∑ₖ₌₀ⁿ rᵏ = (1 - rⁿ⁺¹)/(1 - r), for r ≠ 1

This formula makes geometric series tractable.

Shifting the Index

Sometimes you want to change where a sum starts. This is just relabeling.

∑ₖ₌₁ⁿ aₖ = ∑ⱼ₌₀ⁿ⁻¹ aⱼ₊₁

If you let j = k - 1, then when k = 1, j = 0; when k = n, j = n - 1. The summand aₖ becomes aⱼ₊₁.

The sum is the same—you're just counting differently.

Index shifting is useful when combining sums or matching them to standard forms.

Double Sums

You can sum over multiple indices:

∑ᵢ₌₁³ ∑ⱼ₌₁² aᵢⱼ

This means: for each i from 1 to 3, add up aᵢⱼ for j from 1 to 2.

= (a₁₁ + a₁₂) + (a₂₁ + a₂₂) + (a₃₁ + a₃₂)

Double sums iterate over grids. They're essential in linear algebra (matrix operations) and multivariable calculus.

Key property: If the limits don't depend on each other, you can swap the order:

∑ᵢ₌₁ᵐ ∑ⱼ₌₁ⁿ aᵢⱼ = ∑ⱼ₌₁ⁿ ∑ᵢ₌₁ᵐ aᵢⱼ

Add across rows first, then down columns—or columns first, then across rows. Same answer.

Infinite Series in Sigma Notation

When the upper limit is infinity, we're writing an infinite series:

∑ₖ₌₁^∞ 1/k² = 1 + 1/4 + 1/9 + 1/16 + ...

This is shorthand for "take the limit of partial sums as n → ∞":

∑ₖ₌₁^∞ aₖ = lim_{n→∞} ∑ₖ₌₁ⁿ aₖ

If this limit exists and is finite, the series converges. If not, it diverges.

Sigma notation makes infinite series writable. Without it, we'd need ellipses (...) forever.

Product Notation: Sigma's Sibling

Where ∑ means add, ∏ (capital pi) means multiply:

∏ₖ₌₁⁵ k = 1 × 2 × 3 × 4 × 5 = 120

This is 5! (factorial).

∏ₖ₌₁ⁿ aₖ = a₁ · a₂ · a₃ · ... · aₙ

Product notation shows up in combinatorics, probability, and anywhere multiplication chains appear.

Why Sigma Notation Matters

1. Compression. Writing 100-term sums longhand is impractical. Sigma notation handles any length uniformly.
2. Precision. "1 + 2 + 3 + ..." is ambiguous. ∑ₖ₌₁^∞ k is exact.
3. Manipulation. Sigma notation properties let you transform sums algebraically—pull out constants, split into parts, change indices.
4. Generalization. The same notation handles finite sums, infinite series, and sums over any indexing set.
5. Bridge to calculus. Riemann sums (the foundation of integration) are written in sigma notation. ∫f(x)dx is the limit of ∑f(xₖ)Δx.

Sigma notation is the language of series. It says exactly what to add, from where to where, without ambiguity or tedium.

Reading Tips

When you see a sigma expression:

1. Identify the index. What letter varies?
2. Find the bounds. Where does it start? Where does it stop?
3. Understand the summand. What expression gets evaluated at each step?
4. Write out a few terms. If confused, expand the first 3-4 terms explicitly.

∑ₖ₌₂⁵ (k² - 1) = (4-1) + (9-1) + (16-1) + (25-1) = 3 + 8 + 15 + 24 = 50

The notation is compact but not mysterious. It's just instruction: add these things up.

Part 5 of the Sequences Series series.

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