Sigma Notation
Parts of the notation, a worked example, negative values, and every rule for manipulating a sum
1. The parts of the notation
Upper limit
the last value k takes
5
Sigma
means: add
all of these up
Σ
General term
the formula for a single
term
(2k + 1)
k=1
Index and lower limit
start counting from here
Σ (capital Greek sigma) — the instruction “add these up”. The symbol Σ is the capital Greek letter sigma
and is shorthand for “sum”. (LibreTexts)
Index of summation — the counter (usually k, n, i or j). It takes every integer value between the lower
and upper limits, and each value is substituted into the formula. (Ximera)
Lower limit (below Σ) — where the counter starts. It is the number used to generate the first term of the
series. (Lumen)
Upper limit (above Σ) — where the counter stops. It is the number used to generate the last term. (Lumen)
General term / summand (to the right of Σ) — the rule each term obeys. An explicit formula for each
term is given to the right of the sigma. (Lumen)
One extra fact worth knowing: the index letter is arbitrary. The index variable is a “dummy variable” — it
may be changed to any letter without affecting the value of the summation. (LibreTexts)
6
X
(2n − 1) =
n=3
6
X
6
X
(2k − 1) =
(2j − 1)
j=3
k=3
2. Worked example
Evaluate
5
X
(2k + 1).
k=1
Substitute k = 1, 2, 3, 4, 5 into 2k + 1:
k = 1:
k = 2:
k = 3:
k = 4:
k = 5:
2(1) + 1 = 3
2(2) + 1 = 5
2(3) + 1 = 7
2(4) + 1 = 9
2(5) + 1 = 11
1
5
X
(2k + 1) = 3 + 5 + 7 + 9 + 11 = 35
k=1
Check using the arithmetic series formula Sn =
S5 =
n
2a + (n − 1)d , with a = 3, d = 2, n = 5:
2
5
5
6 + 8 = (14) = 35
2
2
✓
3. Can n be negative?
“n” means different things in different positions, so this splits into four separate answers.
(a) Can the limits be negative? Yes. The lower limit of summation can be any number, but 1 is frequently
used. (Lumen)
2
X
k 2 = (−2)2 + (−1)2 + 02 + 12 + 22 = 4 + 1 + 0 + 1 + 4 = 10
k=−2
(b) Can the number of terms be negative? No. The number of terms is upper − lower + 1, which counts
objects. In one worked example the count is calculated as n = 40 − 18 + 1 = 23. (LibreTexts) A count of
terms is 0 or a positive integer — never negative.
(c) What if the upper limit is smaller than the lower limit? The sum is defined as zero, not negative: if
b < a, then the sum is zero. (Yale) This is called the empty sum.
−5
X
ai = 0
i=0
(d) Can the terms be negative? Yes. “This is a summation. It’s just that we are adding negative values for
the first few terms.” (LibreTexts)
One restriction that is√not about sign: the index moves through integers, in steps of 1 (Ximera). So the
index cannot be 1,5 or 2.
I cannot confirm from a cited source what restrictions the South African NSC / CAPS Grade 12 papers
place on limits in practice — that would need the official CAPS Mathematics document or past papers,
which were not retrieved for this sheet.
2
4. All the ways to manipulate sigma
Group A — the four core properties (all from the same theorem in the LibreTexts / Stitz & Zeager
precalculus text)
1. Split a sum or difference:
2. Take a constant out:
p
X
p
X
p
p
X
X
an ± bn =
an ±
bn
n=m
n=m
p
X
c an = c
n=m
an
n=m
for any real number c
n=m
3. Break the range in two:
p
X
an =
n=m
j
X
an +
n=m
p
X
for m ≤ j < j + 1 ≤ p
an
n=j+1
4. Shift the index (add r to both limits, subtract r inside):
p
X
p+r
X
an =
n=m
an−r
for any whole number r
n=m+r
Group B — other standard moves
5. Rename the index (it is a dummy variable) — see the example in section 1.
6. Sum of a constant:
n
X
C is n copies of C added together, which is nC. (Ximera)
k=1
7. Empty sum is 0: if the upper limit is below the lower limit, the sum is zero. (Yale)
8. Peel off a term:
n
X
ak = a1 +
k=1
n
X
ak — this is rule 3 with j = 1.
k=2
P
9. Factor to force geometric form: manipulate the series into the form b r p , where the power p is 0
when evaluated at the lower limit. (LibreTexts) Worked instance:
1 X
∞
∞
∞ X
3(2)n+1
2
2n
6 X
2 n
=
3
=
−
−
(−5)n+3
(−5)3
(−5)n
125
5
n=0
10. Telescoping:
n
X
n=0
n=0
ak+1 − ak = an+1 − a1 .
k=1
This one is not in the sources listed below — but you can check it by writing it out:
(a2 − a1 ) + (a3 − a2 ) + (a4 − a3 ) + · · · + (an+1 − an )
Every middle term appears once positive and once negative and cancels, leaving an+1 − a1 .
Group C — the closed formulas you evaluate with (all from LibreTexts)
3
11. Powers of the natural numbers:
n
X
k=1
n
X
n(n + 1)
,
k=
2
k=1
12. Arithmetic:
n(n + 1)(2n + 1)
,
k =
6
Sn = n
2
a1 + an
2
=
n
X
k3 =
k=1
n2 (n + 1)2
4
n
2a + (n − 1)d
2
13. Geometric (finite):
Sn = a
1 − rn
1−r
if r ̸= 1,
Sn = na if r = 1
14. Geometric (infinite):
∞
X
ark−1 =
k=1
a
1−r
where |r| < 1;
if |r| ≥ 1 the sum is not defined
5. What you may NOT do
There is no product rule and no quotient rule for sigma. Counterexample you can check yourself:
! 2 !
2
2
X
X
X
(k · k) = 1 + 4 = 5,
but
k
k =3×3=9
k=1
So
X
ak bk ̸=
X
k=1
ak
X
bk , and by the same kind of check
k=1
P
ak
̸= P .
bk
bk
X ak
Sources used
• LibreTexts — Stitz & Zeager, Math 370: Precalculus, §7.2 “Summation Notation” (CC BY-NC-SA, last modified
6 September 2023).
https://math.libretexts.org/Courses/Cosumnes_River_College/Math_370:_Precalculus/07:
_Sequences_and_the_Binomial_Theorem/7.02:_Summation_Notation
• Lumen — Lumen Learning, College Algebra, “Using Summation Notation”.
https://courses.lumenlearning.com/odessa-collegealgebra/chapter/using-summation-notation/
• Ximera — The Ohio State University, “Introduction to sigma notation”.
https://ximera.osu.edu/mooculus/calculus1TextbookBySection/approximatingTheAreaUnderACurve/a
pproximatingTheAreaUnderACurve/digInSigmaNotation
• Yale — J. Aspnes, Yale University Computer Science, “Summation Notation”.
https://cs.yale.edu/homes/aspnes/pinewiki/SummationNotation.html
4