Unit 1B

Limits

Lesson 6: Limits of Sequences and Series

Skill Builder

Infinite Geometric Series

A ball is dropped from a height of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»1«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/mrow»«/mstyle»«/math». When it bounces back up, it reaches a height of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»0«/mn»«mo».«/mo»«mn»5«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/mstyle»«/math». The next bounce reaches «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»0«/mn»«mo».«/mo»«mn»25«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/mstyle»«/math», and so on. The real question is, will it ever stop bouncing? Practically speaking, you will probably say yes because experience has taught you that it will. However, in theory, the ball will never stop bouncing. It will always return to half of its previous height. But, as the number of bounces increases, the subsequent heights reached by the ball get closer and closer to zero.

In other words, as the number of bounces goes to infinity, the heights (the sequence) approach, or have a limit of, zero. In order for an infinite sequence to have a limit, it must be convergent, such as the bouncing ball example. A convergent sequence (or series) has terms that approach a particular value. On the other hand, an infinite sequence will not have a limit if it is divergent. A divergent sequence (or series) has terms that do not approach a particular value. A sum cannot be calculated for infinite divergent geometric series. For infinite geometric series that are convergent, a finite sum can be determined (and there is a formula).

Convergent Series
A series where the terms approach a particular value

Divergent Series
A series where the terms do not approach a particular value

Convergent geometric series have the unique property that the common ratio, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»r«/mi»«/mstyle»«/math», is between «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»-«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math». As the number of terms, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»n«/mi»«/mstyle»«/math», increase indefinitely, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msup»«mi»r«/mi»«mi»n«/mi»«/msup»«/mstyle»«/math» approaches zero when «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»-«/mo»«mn»1«/mn»«mo»§#60;«/mo»«mi»r«/mi»«mo»§#60;«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math». The symbol, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»§#8734;«/mo»«/mstyle»«/math», is a mathematical symbol representing infinity.

Infinite Geometric Series

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msub»«mi»t«/mi»«mn»1«/mn»«/msub»«/mstyle»«/math» is the first term in the series

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»r«/mi»«/mstyle»«/math» is the common ratio «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»-«/mo»«mn»1«/mn»«mo»§#60;«/mo»«mi»r«/mi»«mo»§#60;«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mstyle»«/math»