Sigma Notation
Sigma 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," with the starting index written below and the ending index above. $\sum_{k=m}^{n} a_k$ means add $a_m + a_{m+1} + \ldots + a_n$, where $k$ is the index, $m$ the lower limit, and $n$ the upper limit (which can start at any integer). In analysis, a finite sum $\sum_{k=m}^{n} a_k$ contrasts with an infinite series $\sum_{k=m}^{\infty} a_k$, which requires a convergence condition; the Riemann integral is defined as a limit of Riemann sums, making sigma notation the discrete analog of the integral sign (itself an elongated S for "sum"), and double sums $\sum_i \sum_j a_{ij}$ correspond to iterated integrals by Fubini's theorem.
Example
The sum $1 + 2 + 3 + 4 + 5$ can be written as $\sum_{k=1}^{5} k = 15$. Evaluating $\sum_{k=1}^{4} (2k + 1) = 3+5+7+9 = 24$ illustrates the properties $\sum(a_k + b_k) = \sum a_k + \sum b_k$ and $\sum(c a_k) = c\sum a_k$. 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
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, compressing long sums into one compact symbol; a key point is that the index variable $k$ is a "dummy variable" that disappears in the sum, so $\sum_{k=1}^{n} k^2 = \sum_{j=1}^{n} j^2$. The interchange of sums ($\sum_i \sum_j = \sum_j \sum_i$) requires conditions analogous to Fubini's theorem: absolute convergence guarantees the interchange, while conditional convergence can fail, leading to the Riemann rearrangement theorem.