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← An Introduction to Series

Concept

Sigma Notation

Sigma notation, also known as summation notation, is a compact way of expressing addition. This notation consists of four parts.

The Greek letter sigma $Σ .$ This letter indicates that the terms are added together.
The $general$ $form$ of all the terms being added, in terms of a variable. The variables $n$ and $i$ are commonly used.
The $summation$ $index$ and the $starting$ $index$ , also called the $lower$ $limit .$ They are placed below the letter sigma and are connected with an equals sign. The summation index is the variable used in the general form of the terms. The starting index is the first value that the variable takes.
The $final$ $index$ or $upper$ $limit .$ This number is placed above the letter sigma and indicates the value for the variable in the last term of the summation.

In the example below, all four parts are shown.

The variable

n

— the summation index — only takes integer values. To write this sum explicitly, the variable

n

must be replaced with the integers from the initial value through the final value.

n = 1 \sum 4 2 n n = 1 \sum 4 2 n n = 1 \sum 4 2 n = n = 1 2 (1) + n = 2 2 (2) + n = 3 2 (3) + n = 4 2 (4) = 2 + 4 + 6 + 8 = 20

Extra

Notable Aspects of Summation Notation

There are some aspects that are worth noting.

The summation does not depend on the summation index used.
$n = 1 \sum 4 2 n = k = 1 \sum 4 2 k = i = 1 \sum 4 2 i = 20$
Sometimes a summation may involve other variables. These should not be confused with the summation index.
$n = 1 \sum 3 \frac{2 n}{k}$
Here, the summation index is $n .$ Therefore, the indicated values should only be substituted for $n$ and not for $k .$
$n = 1 \sum 3 \frac{2 n}{k} n = 1 \sum 3 \frac{2 n}{k} n = 1 \sum 3 \frac{2 n}{k} = n = 1 \frac{2 ( 1 )}{k} + n = 2 \frac{2 ( 2 )}{k} + n = 3 \frac{2 ( 3 )}{k} = \frac{2}{k} + \frac{4}{k} + \frac{6}{k} = \frac{1 2}{k}$
The initial value can be any integer less than or equal to the final index. The final index only indicates the last value to be substituted for $n$ — it does not indicate the number of terms.
$n = 8 \sum 10 3 n = n = 8 3 (8) + n = 9 3 (9) + n = 10 3 (10)$

The summation notation is not only useful for working with sums involving a large number of terms, but it can also be used to represent an infinite sum. If an infinite number of terms is to be added, the symbol

\infty

is used in the final index.

n = 1 \sum \infty \frac{1}{2 ^{n}} = \frac{1}{2 ^{1}} + \frac{1}{2 ^{2}} + \frac{1}{2 ^{3}} + \dots

Exercises

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