Khanh at 02:20, 15 November 2021

2021-11-15T02:20:13Z

Lila: /* Common summations */

2021-10-19T18:43:17Z

Common summations

Lila: Created page with "Summation notation allows an expression that contains a sum to be expressed in a simple, compact manner. The uppercase Greek letter sigma, Σ, is used to denote the sum o..."

2021-10-19T18:42:40Z

Created page with "Summation notation allows an expression that contains a sum to be expressed in a simple, compact manner. The uppercase Greek letter sigma, Σ, is used to denote the sum o..."

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Summation notation allows an expression that contains a sum to be expressed in a simple, compact manner. The uppercase Greek letter sigma, Σ, is used to denote the sum of a set of numbers.

;Example
:<math>\sum_{i=3}^7 i^2=3^2+4^2+5^2+6^2+7^2</math>

<blockquote style="background: white; border: 1px solid black; padding: 1em;">
Let <math>f</math> be a function and <math>M,N</math> are integers with <math>N<M</math> . Then
:<math>\sum_{i=N}^M f(i)=f(N)+f(N+1)+f(N+2)+\cdots+f(M)</math> .
We say <math>N</math> is the lower limit and <math>M</math> is the upper limit of the sum.
</blockquote>

We can replace the letter <math>i</math> with any other variable. For this reason <math>i</math> is referred to as a ''dummy variable''. So
:<math>\sum_{i=1}^4 i=\sum_{j=1}^4 j=\sum_{\alpha=1}^4 \alpha=1+2+3+4</math>
Conventionally we use the letters <math>i</math> , <math>j</math> , <math>k</math> , <math>m</math> for dummy variables.

;Example
:<math>\sum_{i=1}^5 i=1+2+3+4+5</math>

Here, the ''dummy variable'' is <math>i</math> , the ''lower limit'' of summation is 1, and the ''upper limit'' is 5.

;Example

Sometimes, you will see summation signs with no dummy variable specified, e.g.,

:<math>\sum_1^4 i^3=100</math>

In such cases the correct dummy variable should be clear from the context.

You may also see cases where the limits are unspecified. Here too, they must be deduced from the context.

===Common summations===
{{Wikipedia|Summation#Capital-sigma notation}}

<math>\sum_{i=1}^n c=c+c+\cdots+c=nc\ ,\ c\in\R</math>

<math>\sum_{i=1}^n i=1+2+3+\cdots+n=\frac{n(n+1)}{2}</math>

<math>\sum_{i=1}^n i^2=1^2+2^2+3^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}</math>

<math>\sum_{i=1}^n i^3=1^3+2^3+3^3+\cdots+n^3=\left(\sum_{i=1}^n i\right)^2=\left(\frac{n(n+1)}{2}\right)^2</math>

<math>\sum_{i=1}^\infty a_i = \lim_{t \to \infty} \left[\sum_{i=1}^t a_i\right]</math>

==Resources==
* [https://en.wikibooks.org/wiki/Calculus/Summation_notation Summation notation], Wikibooks: Calculus

← Older revision		Revision as of 18:43, 19 October 2021
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	===Common summations===		===Common summations===
−	~~{{Wikipedia\|Summation#Capital-sigma notation}}~~

	<math>\sum_{i=1}^n c=c+c+\cdots+c=nc\ ,\ c\in\R</math>		<math>\sum_{i=1}^n c=c+c+\cdots+c=nc\ ,\ c\in\R</math>

@@ Line 41: / Line 41: @@
 <math>\sum_{i=1}^\infty a_i = \lim_{t \to \infty} \left[\sum_{i=1}^t a_i\right]</math>
-==Resources==
+== Licensing ==
-* [https://en.wikibooks.org/wiki/Calculus/Summation_notation Summation notation], Wikibooks: Calculus
+Content obtained and/or adapted from:

Sigma Notation - Revision history

Khanh at 02:20, 15 November 2021

Lila: /* Common summations */

Lila: Created page with "Summation notation allows an expression that contains a sum to be expressed in a simple, compact manner. The uppercase Greek letter sigma, Σ, is used to denote the sum o..."