Sums

Overview

The summation of a sequence of terms is denoted and defined as follows:

\sum_{k = m}^{n} a_{k} = a_{m} + a_{m + 1} + \dots + a_{n}

The resulting number is called the sum. Variable $k$ is called the summation index. It ranges from the start index to the end index.

If $a_{1}, a_{2}, \dots, a_{n}$ are $n$ real numbers, their arithmetic mean $\bar{a}$ is defined by the equation

\bar{a} = \frac{1}{n} \sum_{k = 1}^{n} a_{k} .

If $w_{1}, w_{2}, \dots, w_{n}$ are nonnegative real numbers, not all zero, the weighted arithmetic mean $\bar{a}$ is defined by the formula

\bar{a} = \frac{\sum_{k = 1}^{n} w_{k} a_{k}}{\sum_{k = 1}^{n} w_{k}} .

In this context, $w_{1}, w_{2}, \dots, w_{n}$ are called weights.

Let $(a_{n})$ and $(b_{n})$ be sequences. Then

\begin{aligned} \sum_{k = m}^{n} a_{k} Δ b_{k} & = (a_{n + 1} b_{n + 1} - a_{m} b_{m}) - \sum_{k = m}^{n} b_{k + 1} Δ a_{k} \\ \sum_{k = m}^{n} a_{k} \nabla b_{k} & = (a_{n} b_{n} - a_{m - 1} b_{m - 1}) - \sum_{k = m}^{n} b_{k - 1} \nabla a_{k}, \end{aligned}

where $Δ$ and $\nabla$ refer to difference operators. These formulas are the discrete analogue to integration by parts.