Sigma Notation

Sigma notation packs an entire sum into one line: a index of summation runs from a lower limit to an upper limit, and you plug each integer into a formula and add the results.

By the end you'll be able to expand a sum written in sigma notation into its terms and add them, and recognize when the rule inside is linear — making the sum an arithmetic series in disguise.

Predict: what is \(\displaystyle\sum_{i=1}^{4} i\)? Expand the terms below and press Play to check.

Set the lower limit, upper limit, and rule, then watch each term expand and accumulate into the running sum. The index i is color-linked between the formula and every expanded term.

\(\displaystyle\sum_{i=1}^{4} i\)

i=11
i=22
i=33
i=44

Running sum: 10 (4 terms added)

index i term being added

Sigma notation is shorthand for a sum: substitute each integer index into a formula and add up the results — the same three ingredients every time, a start, an end, and a rule.

Formal

\(\displaystyle\sum_{i=\text{start}}^{n} f(i) = f(\text{start}) + f(\text{start}+1) + \cdots + f(n)\). The index i is a dummy variable — renaming it (to k, j, or anything else) changes nothing about the value, since it never appears in the final answer. Two useful rules follow directly from the definition:

  • Constant factor: \(\displaystyle\sum c\cdot f(i) = c\cdot\sum f(i)\) — pull a constant outside the sum.
  • Sum splits: \(\displaystyle\sum \big(f(i)+g(i)\big) = \sum f(i) + \sum g(i)\) — add two sums term by term.
Procedural

To expand any sigma expression: (1) list every integer from the lower limit to the upper limit — there are (upper − lower + 1) of them, not (upper − lower); (2) plug each one into the formula; (3) add the results. If the formula is linear in i, like \(2i+3\), the terms it generates are exactly an arithmetic sequence — so the sum is the arithmetic series formula \(S_n = n(a_1+a_n)/2\) in disguise. There's no need to add term by term once you recognize that pattern.

Applied

Sigma notation is how you'd write "add up this month's sales for every day from the 1st to the 30th" without listing 30 separate numbers — a formula for day i (like a fixed amount per day, or growing sales) plus a start and end day is all you need to describe the whole total compactly.

Worked example

Evaluate \(\displaystyle\sum_{k=1}^{5} (2k+1)\). List k = 1, 2, 3, 4, 5 and plug each into \(2k+1\): \(k{=}1{:}\,3\), \(k{=}2{:}\,5\), \(k{=}3{:}\,7\), \(k{=}4{:}\,9\), \(k{=}5{:}\,11\). Add: \(3+5+7+9+11 = \) 35. (These terms form an arithmetic series with \(d=2\), so \(S_5 = 5(3+11)/2 = 35\) confirms it.)

Your turn

Evaluate \(\displaystyle\sum_{j=1}^{4} (3j-1)\). List j = 1, 2, 3, 4 and plug each into \(3j-1\): \(j{=}1{:}\,2\), \(j{=}2{:}\,5\), \(j{=}3{:}\,8\), \(j{=}4{:}\)____. Now add all four terms: \(2+5+8+\)____ \(=\) ____

Reveal the answer

\(j=4\): \(3(4)-1=11\). Adding: \(2+5+8+11=\) 26. Set the lower limit to 1, upper limit to 4, and rule to "3i + 2" on the builder above (the closest match) to see a similar linear expansion animate term by term.

More info — why a linear rule gives an arithmetic series

If \(f(i) = mi + b\), then consecutive terms \(f(i+1) - f(i) = m\) always differ by the same constant \(m\) — that's exactly the definition of a common difference \(d\) in an arithmetic sequence. So any \(\displaystyle\sum_{i=1}^{n}(mi+b)\) is just the sum of an arithmetic sequence with \(a_1 = m+b\) and \(d=m\), and you can skip expanding every term and jump straight to \(S_n = n(a_1+a_n)/2\). See the Paul's Online Notes link in Dive deeper for the constant-factor and sum-splitting rules used to prove this.

Check your understanding

Question 1 of 4

Evaluate \(\sum_{i=1}^{4} 3i\).

Question 2 of 4

How many terms does \(\sum_{i=2}^{7} f(i)\) expand into?

Question 3 of 4

Which expression is equal to \(\sum_{i=1}^{3} i^2\)?

Question 4 of 4

\(\sum_{i=1}^{10} (2i+3)\) expands into terms that form an arithmetic sequence with \(a_1=5\) and common difference \(d=2\), so \(a_{10}=23\). Using \(S_n = n(a_1+a_n)/2\), what does the sum equal?

Recap

  • \(\displaystyle\sum_{i=\text{start}}^{n} f(i)\) means: plug every integer from the lower limit to the upper limit into f, and add the results.
  • There are (upper − lower + 1) terms — count carefully, both endpoints are included.
  • The index of summation is a dummy variable; renaming it changes nothing.
  • Constant factor: \(\sum c\cdot f(i) = c\cdot\sum f(i)\). Sum splits: \(\sum(f(i)+g(i)) = \sum f(i)+\sum g(i)\).
  • A linear rule \(f(i)=mi+b\) generates an arithmetic sequence, so its sum can be found directly with \(S_n=n(a_1+a_n)/2\) instead of adding term by term.

Dive deeper

Sources

  • Sigma (Summation) Notation