Sigma Notation
Every time a spreadsheet totals a column, a shop tallies a month of takings, or a scientist
averages a thousand readings, the same thing is happening underneath: a long list of numbers
is being added up. Mathematics needed a compact way to say "add all of these," and that is
what sigma notation gives you.
Add up every whole number from 1 to 100. Written out in full,
1 + 2 + 3 + \dots + 98 + 99 + 100 is a mouthful — and that is a
short sum. Real mathematics is full of sums with hundreds of terms, or a number of terms
that depends on some n you haven't chosen yet. Writing them all out is
hopeless.
So mathematicians invented a single symbol that means "add all of these up": the Greek capital
sigma, \sum. Give it a formula, a start value, and an end
value, and it packs an entire sum into one tidy instruction:
\sum_{k=1}^{100} k \;=\; 1 + 2 + 3 + \dots + 100.
That compact bundle on the left says exactly the same thing as the long trail on the right — it is
just a far better way to write it, and to reason about it.
Reading the recipe
Sigma notation is a recipe with four parts. Take a small example:
\sum_{k=1}^{5} k \;=\; 1 + 2 + 3 + 4 + 5 \;=\; 15.
The letter k is the index (a counter). It starts at the
number below the sigma — the lower limit — climbs by one each time, and stops at the
number on top — the upper limit. The expression after the sigma is the
general term (or summand): substitute each value of
k into it and add all the results.
The general term can be any rule at all. Whatever it is, you substitute
k = 1, 2, 3, \dots up to the top, then total the terms:
\sum_{k=1}^{4} 2k \;=\; 2 + 4 + 6 + 8 \;=\; 20,
\sum_{k=1}^{3} k^2 \;=\; 1 + 4 + 9 \;=\; 14.
The lower limit need not be 1. In
\sum_{k=2}^{4} k^2 = 4 + 9 + 16 = 29 the counter runs from
2 to 4, so there are three terms. This is
exactly how an
arithmetic or geometric series
is written compactly — the general term is just the sequence's
nth-term rule.
In \displaystyle\sum_{k=1}^{n} a_k:
- \sum means add the terms;
- k is the index, running from the bottom value
up to the top value, one step at a time;
- a_k is the general term — substitute each
k into it;
- the number of terms is (\text{upper}) - (\text{lower}) + 1.
Watch a sigma unpack
Step through \displaystyle\sum_{k=1}^{4} k^2: substitute each index value,
square it, and total. Notice the answer collapses to a single number.
Worked example — sum of a linear rule
Evaluate \displaystyle\sum_{r=1}^{4} (3r - 1). (The index is called
r here — the letter doesn't matter.) Substitute
r = 1, 2, 3, 4 into 3r - 1:
(3\cdot 1 - 1) + (3\cdot 2 - 1) + (3\cdot 3 - 1) + (3\cdot 4 - 1) = 2 + 5 + 8 + 11 = 26.
Four terms (from 4 - 1 + 1 = 4), each worked out and added. The result is
just the number 26 — no r left in sight.
Two results worth memorising
Some sums come up so often they have closed formulas — no need to add term by term. The simplest is a
sum of a constant. If every one of n terms equals
1, the total is just n:
\sum_{k=1}^{n} 1 = n.
The other is the sum of the first n counting numbers, the very
1 + 2 + \dots + n we opened with. A schoolboy Gauss famously spotted its
formula in seconds:
\sum_{k=1}^{n} k = \frac{n(n+1)}{2}.
Try it on our opening challenge: \sum_{k=1}^{100} k = \frac{100\cdot 101}{2} = 5050
— all one hundred numbers added, with no adding at all.
Splitting and scaling a sum
Because a sigma is just an addition in disguise, it obeys the ordinary rules of adding. Two shortcuts
make sums much easier to handle. First, a constant multiplier can be pulled out to the
front — since it multiplies every single term:
\sum_{k=1}^{n} c\,a_k \;=\; c \sum_{k=1}^{n} a_k.
Second, a sum of two rules splits into two separate sums (you can add the terms in
any order you like):
\sum_{k=1}^{n} (a_k + b_k) \;=\; \sum_{k=1}^{n} a_k \;+\; \sum_{k=1}^{n} b_k.
Together these turn a scary-looking sum into pieces you already know. For instance, using the two
results above,
\sum_{k=1}^{n} (2k + 3) \;=\; 2\sum_{k=1}^{n} k \;+\; \sum_{k=1}^{n} 3 \;=\; 2\cdot\frac{n(n+1)}{2} + 3n \;=\; n(n+1) + 3n \;=\; n^2 + 4n.
No term-by-term slog — just split, scale, and slot in the standard formulas.
The letter under the sigma (the index, usually i,
r, or k) is a dummy variable.
It is not an unknown you solve for — it is just a counter that ticks through the terms and then
vanishes. So \sum_{k=1}^{3} k^2 and
\sum_{i=1}^{3} i^2 are the same sum; renaming the counter changes
nothing. And the finished value is a single number (or a formula in
n) — never an expression that still contains
k. If a k survives in your answer, you forgot to
substitute.
The other classic slip is an off-by-one in the limits. Always count the terms with
(\text{upper}) - (\text{lower}) + 1. Watch:
\sum_{i=1}^{n} i \;\ne\; \sum_{i=0}^{n} i.
The second one has an extra i = 0 term at the front — that's
n+1 terms, not n (though here the values happen
to total the same, since the extra term is 0). Starting at
0 versus 1 almost always changes the count, so
check the bottom limit every time.
Sigma notation is the language of series, and it runs right through higher mathematics and
physics:
- It writes the arithmetic and geometric series
in one line each, formulas and all.
- It writes the Taylor series — the infinite sums like
e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!} that let a calculator actually
compute e^x, \sin x and
\cos x from nothing but additions and multiplications.
- It captures the eerie infinite sums, where letting the upper limit run off to
\infty can still give a finite answer:
\sum_{k=1}^{\infty} \frac{1}{2^k} = \tfrac12 + \tfrac14 + \tfrac18 + \dots = 1.
A never-ending pile of numbers that adds up to exactly one — the first strange, wonderful hint of
how infinity behaves.
Compact, precise, and absolutely everywhere in university maths. Learn to read
\sum now and a whole library opens up.
See it explained