Summation and Product Notation¶

The Summation Sign Σ¶

The summation sign $\Sigma$ (Greek: Sigma) expresses the addition of multiple terms in compact form:

\[ \sum_{k=1}^{n} a_k = a_1 + a_2 + \cdots + a_n \]

Here $k$ is the index of summation, $1$ is the lower bound, and $n$ is the upper bound. The expression $a_k$ is the summand.

Example. $$ \sum_{k=1}^{4} k^2 = 1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30 $$

Linearity: $$ \sum_{k=1}^{n} (c \cdot a_k + b_k) = c \cdot \sum_{k=1}^{n} a_k + \sum_{k=1}^{n} b_k $$
Constant summands: $$ \sum_{k=1}^{n} c = n \cdot c $$
Index shift: Under the substitution $j = k - 1$, the sum $\sum_{k=1}^{n} a_k$ becomes $\sum_{j=0}^{n-1} a_{j+1}$. The value does not change.

The product sign $\Pi$ (Greek: Pi) expresses the multiplication of multiple factors:

\[ \prod_{k=1}^{n} a_k = a_1 \cdot a_2 \cdots a_n \]

Example. The factorial can be written as a product: $$ n! = \prod_{k=1}^{n} k = 1 \cdot 2 \cdot 3 \cdots n $$

Exponentiation: $$ \prod_{k=1}^{n} c = c^n $$
Product of powers: $$ \prod_{k=1}^{n} a_k^{m} = \left(\prod_{k=1}^{n} a_k\right)^{m} $$

The index need not start at $1$ or increase in integer steps. The general form is:

\[ \sum_{k \in I} a_k \]

Here $I$ is a finite index set.

Example. Sum over all primes up to $10$: $$ \sum_{p \in {2,3,5,7}} \frac{1}{p} = \frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} $$

With two running indices, a double sum arises:

\[ \sum_{i=1}^{m} \sum_{j=1}^{n} a_{ij} \]

For finite sums, the order of summation is interchangeable:

\[ \sum_{i=1}^{m} \sum_{j=1}^{n} a_{ij} = \sum_{j=1}^{n} \sum_{i=1}^{m} a_{ij} \]

Example. $$ \sum_{i=1}^{2} \sum_{j=1}^{3} ij = \sum_{i=1}^{2} (i \cdot 1 + i \cdot 2 + i \cdot 3) = \sum_{i=1}^{2} 6i = 6 + 12 = 18 $$

The following formulas appear frequently in number theory:

Arithmetic sum (Gauss): $$ \sum_{k=1}^{n} k = \frac{n(n+1)}{2} $$

Geometric sum (for $q \neq 1$): $$ \sum_{k=0}^{n} q^k = \frac{q^{n+1} - 1}{q - 1} $$

Geometric series (for $|q| < 1$): $$ \sum_{k=0}^{\infty} q^k = \frac{1}{1-q} $$

"The notation $\sum$ for summation was introduced by Euler in 1755." — Florian Cajori, A History of Mathematical Notations, Dover, 1993.

A central example from number theory connects summation and product notation. Euler showed:

\[ \sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}} \quad (s > 1) \]

The left side is an infinite series (the Riemann zeta function), the right side an infinite product over all primes.

Cajori, Florian: A History of Mathematical Notations. Dover, 1993. Volume 2, §§ 438–439.
Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren: Concrete Mathematics. Addison-Wesley, 2nd edition, 1994. Chapter 2.