Sigma Notation
Functions & Advanced AlgebraSigma notation uses the Greek letter sigma to compactly represent a sum of many terms according to a pattern.
Formula
\sum_{k=1}^{n} a_k
Definition
Sigma notation is a shorthand way to write a sum. The Greek capital letter sigma (the symbol that looks like a big E) means "add up all of these." Below the sigma is the starting index; above is the ending index.
Example
The sum $1 + 2 + 3 + 4 + 5$ can be written as $\sum_{k=1}^{5} k$. Plug in $k=1, 2, 3, 4, 5$ and add: $1+2+3+4+5 = 15$.
Key Insight
Sigma notation is like a loop instruction: "start at this value, end at that value, evaluate the expression at each step, and add everything up." It compresses long sums into one compact symbol.
Definition
Sigma notation $\sum_{k=m}^{n} a_k$ means add $a_m + a_{m+1} + \ldots + a_n$. The variable $k$ is the index, $m$ is the lower limit, $n$ is the upper limit. The index can start at any integer.
Example
$\sum_{k=1}^{4} (2k + 1) = (2(1)+1) + (2(2)+1) + (2(3)+1) + (2(4)+1) = 3+5+7+9 = 24$. Properties: $\sum(a_k + b_k) = \sum a_k + \sum b_k$; $\sum(c a_k) = c\sum a_k$.
Key Insight
Sigma notation cleanly expresses series of any length. Knowing the index variable is a "dummy variable" (it disappears in the sum) is key: $\sum_{k=1}^{n} k^2 = \sum_{j=1}^{n} j^2$.
Definition
In analysis, $\sum_{k=m}^{n} a_k$ is a finite sum, while $\sum_{k=m}^{\infty} a_k$ is an infinite series requiring a convergence condition. The Riemann integral is defined as a limit of Riemann sums, making sigma notation the discrete analog of the integral sign (which is itself an elongated S for "sum"). Double sums $\sum_i \sum_j a_{ij}$ correspond to iterated integrals by Fubini's theorem.
Example
The Riemann zeta function: $\zeta(s) = \sum_{n=1}^{\infty} 1/n^s$, converges for $\text{Re}(s) > 1$. At $s=2$: $\zeta(2) = \pi^2/6$, proved by Euler using the Fourier series for $x^2$.
Key Insight
The interchange of sums ($\sum_i \sum_j = \sum_j \sum_i$) requires conditions analogous to Fubini's theorem. Absolute convergence guarantees the interchange; conditional convergence can fail, leading to the Riemann rearrangement theorem.