L6 Limits of Sequences and Series - Part 2
Completion requirements
Unit 1B
Limits
Lesson 6: Limits of Sequences and Series
Determine the general term for the sequence «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#8722;«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«mi
mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8722;«/mo»«mfrac»«mn»1«/mn»«mn»9«/mn»«/mfrac»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mfrac»«mn»1«/mn»«mn»16«/mn»«/mfrac»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8722;«/mo»«mfrac»«mn»1«/mn»«mn»25«/mn»«/mfrac»«mo»,«/mo»«mo»§#160;«/mo»«mo»...«/mo»«/mrow»«/mstyle»«/math»
The general term of this sequence requires consideration of two components.
Combining these two components, the general term is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»t«/mi»«mi»n«/mi»«/msub»«mo»=«/mo»«msup»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mi»n«/mi»«/msup»«mfrac»«mn»1«/mn»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«/mfrac»«/mrow»«/mstyle»«/math».
Limits of Sequences
Although infinite sequences continue indefinitely, the general term, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msub»«mi»t«/mi»«mi»n«/mi»«/msub»«/mstyle»«/math», can approach a specific number, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»L«/mi»«/mstyle»«/math», as the number of terms approaches infinity. This can be represented by a limit.
Consider the sequence formed by the general term «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»t«/mi»«mi»n«/mi»«/msub»«mo»=«/mo»«mfrac»«mn»1«/mn»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«/mfrac»«/mrow»«/mstyle»«/math».
The first five terms of the sequence are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mfrac»«mn»1«/mn»«mn»9«/mn»«/mfrac»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mfrac»«mn»1«/mn»«mn»16«/mn»«/mfrac»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mfrac»«mn»1«/mn»«mn»25«/mn»«/mfrac»«/mrow»«/mstyle»«/math».
Graphing these terms helps to visualize the behaviour of the sequence function.
As the term number, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»n«/mi»«/mstyle»«/math», gets larger, the term gets closer and closer to zero. This is the called the limit of the sequence.
For «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»t«/mi»«mi»n«/mi»«/msub»«mo»=«/mo»«mfrac»«mn»1«/mn»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«/mfrac»«/mrow»«/mstyle»«/math», the limit of the sequence is as follows.
Compare this sequence to the sequence formed by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»t«/mi»«mi»n«/mi»«/msub»«mo»=«/mo»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math». Graph the first five terms, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»4«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»9«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»16«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»25«/mn»«/mrow»«/mstyle»«/math», to visualize the behaviour of the sequence function.
Unlike the sequence defined by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»t«/mi»«mi»n«/mi»«/msub»«mo»=«/mo»«mfrac»«mn»1«/mn»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«/mfrac»«/mrow»«/mstyle»«/math», this sequence, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»t«/mi»«mi»n«/mi»«/msub»«mo»=«/mo»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math», keeps increasing towards infinity. It does not approach a specific number. Therefore, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«munder»«mi»lim«/mi»«mrow»«mi»n«/mi»«mo»§#8594;«/mo»«mo»§#8734;«/mo»«/mrow»«/munder»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math» does not exist.
If the terms of an infinite sequence approach a unique finite value, the sequence is called a convergent sequence. A sequence non-convergent sequence does not approach a specific number and is called a divergent sequence. An example of a convergent sequence is produced by the general term «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»t«/mi»«mi»n«/mi»«/msub»«mo»=«/mo»«mfrac»«mn»1«/mn»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«/mfrac»«/mrow»«/mstyle»«/math», whereas an example of a divergent sequence is produced by the general term «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»t«/mi»«mi»n«/mi»«/msub»«mo»=«/mo»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math».
i.
The denominators are increasing. Each is the square of the term number. This can be represented by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»t«/mi»«mi»n«/mi»«/msub»«mo»=«/mo»«mfrac»«mn»1«/mn»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«/mfrac»«/mrow»«/mstyle»«/math».
ii.
The terms have alternating signs. This can be represented by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»t«/mi»«mi»n«/mi»«/msub»«mo»=«/mo»«msup»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mi»n«/mi»«/msup»«/mrow»«/mstyle»«/math».
Combining these two components, the general term is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»t«/mi»«mi»n«/mi»«/msub»«mo»=«/mo»«msup»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mi»n«/mi»«/msup»«mfrac»«mn»1«/mn»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«/mfrac»«/mrow»«/mstyle»«/math».
Limits of Sequences
Although infinite sequences continue indefinitely, the general term, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msub»«mi»t«/mi»«mi»n«/mi»«/msub»«/mstyle»«/math», can approach a specific number, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»L«/mi»«/mstyle»«/math», as the number of terms approaches infinity. This can be represented by a limit.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«munder»«mi»lim«/mi»«mrow»«mi»n«/mi»«mo»§#8594;«/mo»«mo»§#8734;«/mo»«/mrow»«/munder»«msub»«mi»t«/mi»«mi»n«/mi»«/msub»«mo»=«/mo»«mi»L«/mi»«/mrow»«/mstyle»«/math»
Consider the sequence formed by the general term «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»t«/mi»«mi»n«/mi»«/msub»«mo»=«/mo»«mfrac»«mn»1«/mn»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«/mfrac»«/mrow»«/mstyle»«/math».
The first five terms of the sequence are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mfrac»«mn»1«/mn»«mn»9«/mn»«/mfrac»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mfrac»«mn»1«/mn»«mn»16«/mn»«/mfrac»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mfrac»«mn»1«/mn»«mn»25«/mn»«/mfrac»«/mrow»«/mstyle»«/math».
Graphing these terms helps to visualize the behaviour of the sequence function.
As the term number, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»n«/mi»«/mstyle»«/math», gets larger, the term gets closer and closer to zero. This is the called the limit of the sequence.
For «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»t«/mi»«mi»n«/mi»«/msub»«mo»=«/mo»«mfrac»«mn»1«/mn»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«/mfrac»«/mrow»«/mstyle»«/math», the limit of the sequence is as follows.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«munder»«mi»lim«/mi»«mrow»«mi»n«/mi»«mo»§#8594;«/mo»«mo»§#8734;«/mo»«/mrow»«/munder»«mfrac»«mn»1«/mn»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«/mfrac»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math»
Compare this sequence to the sequence formed by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»t«/mi»«mi»n«/mi»«/msub»«mo»=«/mo»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math». Graph the first five terms, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»4«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»9«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»16«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»25«/mn»«/mrow»«/mstyle»«/math», to visualize the behaviour of the sequence function.
Unlike the sequence defined by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»t«/mi»«mi»n«/mi»«/msub»«mo»=«/mo»«mfrac»«mn»1«/mn»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«/mfrac»«/mrow»«/mstyle»«/math», this sequence, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»t«/mi»«mi»n«/mi»«/msub»«mo»=«/mo»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math», keeps increasing towards infinity. It does not approach a specific number. Therefore, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«munder»«mi»lim«/mi»«mrow»«mi»n«/mi»«mo»§#8594;«/mo»«mo»§#8734;«/mo»«/mrow»«/munder»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math» does not exist.
If the terms of an infinite sequence approach a unique finite value, the sequence is called a convergent sequence. A sequence non-convergent sequence does not approach a specific number and is called a divergent sequence. An example of a convergent sequence is produced by the general term «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»t«/mi»«mi»n«/mi»«/msub»«mo»=«/mo»«mfrac»«mn»1«/mn»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«/mfrac»«/mrow»«/mstyle»«/math», whereas an example of a divergent sequence is produced by the general term «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»t«/mi»«mi»n«/mi»«/msub»«mo»=«/mo»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math».
It is important to note that as soon as a function is defined as a sequence, it is implied the domain is the set of natural numbers, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»1«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»2«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»3«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mo»§#8230;«/mo»«/mrow»«/mstyle»«/math».
Determine if the sequence «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»n«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»4«/mn»«mo»§#8722;«/mo»«mi»n«/mi»«/mrow»«/mstyle»«/math»
converges or diverges by graphing the first «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»6«/mn»«/mstyle»«/math» terms of the sequence.
Set up a table of values to determine the values of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»n«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»4«/mn»«mo»§#8722;«/mo»«mi»n«/mi»«/mrow»«/mstyle»«/math»
for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»n«/mi»«/mstyle»«/math»-values of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math»
to «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»6«/mn»«/mstyle»«/math». Graph the results.
This is a divergent sequence because the sequence continues to decrease at a constant rate and does not approach a specific number.
| «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»n«/mi»«/mstyle»«/math» | «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»n«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»4«/mn»«mo»§#8722;«/mo»«mi»n«/mi»«/mrow»«/mstyle»«/math» |
| «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math» | «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«/mstyle»«/math» |
| «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math» | «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math» |
| «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«/mstyle»«/math» | «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math» |
| «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»4«/mn»«/mstyle»«/math» | «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»0«/mn»«/mstyle»«/math» |
| «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»5«/mn»«/mstyle»«/math» | «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»-«/mo»«mn»1«/mn»«/mstyle»«/math» |
| «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»6«/mn»«/mstyle»«/math» | «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»-«/mo»«mn»2«/mn»«/mstyle»«/math» |
This is a divergent sequence because the sequence continues to decrease at a constant rate and does not approach a specific number.