Summation is the operation of adding a sequence of numbers, resulting in a sum or total. If numbers are added sequentially from left to right, any intermediate result is a partial sum. The numbers to be summed (called addends, or sometimes summands) may be integers, rational numbers, real numbers, or complex numbers. For finite sequences of such elements, summation always produces a well-defined sum.

A series is a list of numbers—like a sequence—but instead of just listing them, the plus signs indicate that they should be added up. 

For example, $4+9+3+2+17$ is a series. This particular series adds up to $35$. Another series is $2+4+8+16+32+64$. This series sums to $126$.

Sigma Notation

One way to compactly represent a series is with sigma notation, or summation notation, which looks like this:

$\displaystyle{\sum _{n=3}^{7}{n^2}}$

The main symbol seen is the uppercase Greek letter sigma. It indicates a series. To "unpack" this notation, $n=3$ represents the number at which to start counting ($3$), and the $7$ represents the point at which to stop. For each term, plug that value of $n$ into the given formula ($n^2$). This particular formula, which we can read as "the sum as $n$ goes from $3$ to $7$ of $n^2$," means:

$\displaystyle{3^2 + 4^2 + 5^2 + 6^2 + 7^2}$

More generally, sigma notation can be defined as:

$\displaystyle{\sum _{ i=m }^{ n }{ x_i }=x_m+x_{m+1}+x_{m+2}+...+x_{n-1}+x_n}$

In this formula, i represents the index of summation, $x_i$ is an indexed variable representing each successive term in the series, $m$ is the lower bound of summation, and $n$ is the upper bound of summation. The "$i = m$" under the summation symbol means that the index $i$ starts out equal to $m$. The index, $i$, is incremented by $1$ for each successive term, stopping when $i=n$.

Another example is:

$\displaystyle{ \begin{aligned} \sum_{i=3}^6 (i^2+1) &= (3^2+1)+(4^2+1)+(5^2+1)+(6^2+1) \\ &=10+17+26+37 \\ &=90 \end{aligned} }$ 

This series sums to $90.$ So we could write:

$\displaystyle \sum_{i=3}^6 (i^2+1)=90$

Other Forms of Sigma Notation

Informal writing sometimes omits the definition of the index and bounds of summation when these are clear from context. For example:

$\displaystyle{\sum x_i^2=\sum _{ i=1 }^n x_i^2}$