Unit 7A

Integrals Part 1

Lesson 3: Areas Part 1

Skill Builder

Summation Notation

A series is the sum of the terms in a sequence. If «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»t«/mi»«mi»n«/mi»«/msub»«/math» is the general term of a sequence, the sum of the first «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»n«/mi»«/math» terms of that sequence is given by the series «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»S«/mi»«mi»n«/mi»«/msub»«/math».


The Greek letter «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8721;«/mo»«mspace/»«/math» (sigma) stands for sum. Sigma notation (also called summation notation) is often used to abbreviate the writing of a series. For example, the series formed by adding up the first «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»4«/mn»«/math» terms of the sequence defined by «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»t«/mi»«mi»i«/mi»«/msub»«mo»=«/mo»«mn»3«/mn»«mi»i«/mi»«/math» is written as follows.
 

The values of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»i«/mi»«/math», from «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»i«/mi»«mo»=«/mo»«mn»1«/mn»«/math» to «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»i«/mi»«mo»=«/mo»«mn»4«/mn»«/math», are substituted into the general term of the sequence «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»t«/mi»«mi»i«/mi»«/msub»«mo»=«/mo»«mn»3«/mn»«mi»i«/mi»«/math» to find the sum of the series.


The variable «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»i«/mi»«/math» is called the index of summation. The number below «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8721;«/mo»«mspace/»«/math» is the lower limit of summation and the number above «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8721;«/mo»«mspace/»«/math» is the upper limitof summation. The number of terms in a series expressed in summation notation is found as follows.


For the example above, the number of terms is as follows.


In the expanded form of the series above, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»4«/mn»«/math» terms were added together to find the sum.