L6 Limits of Sequences and Series - Part 1
Completion requirements
Unit 1B
Limits
Lesson 6: Limits of Sequences and Series
How would you calculate the area of a circle if you didn’t know the formula «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»A«/mi»«mo»=«/mo»«mo»§#960;«/mo»«msup»«mi»r«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math»?
In the 3rd century AD, Liu Hui did this by inscribing polygons inside a circle, and then determining their areas. The concept is remarkably simple. By increasing the number of vertices on the polygon, the area becomes a better and better approximation
of the actual area of the circle.
Click the interactive button to allow you to visualize what Liu Hui did.
As the number of sides on the inscribed polygon increases, the area of the polygon increases. However, it does not increase at the same rate with every additional side. In fact, as more and more sides are added, the areas of the polygons begin to converge on, or approach, a definite value, which is in fact the actual area of the circle!
Interactive
Click the interactive button to allow you to visualize what Liu Hui did.
As the number of sides on the inscribed polygon increases, the area of the polygon increases. However, it does not increase at the same rate with every additional side. In fact, as more and more sides are added, the areas of the polygons begin to converge on, or approach, a definite value, which is in fact the actual area of the circle!
To determine the area that the inscribed polygons approach, use the Applet to fill in the table below.
| Number of Sides
|
«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¨»«mn»3«/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»5«/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¨»«mn»7«/mn»«/mstyle»«/math» |
| Area of Inscribed Polygon | «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»f«/mi»«mfenced»«mi»n«/mi»«/mfenced»«/mstyle»«/math»
|
Click here to view the populated table.
A sequence is a list of numbers separated by commas given in a definite order. A sequence can be expressed as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»t«/mi»«mn»1«/mn»«/msub»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«msub»«mi»t«/mi»«mn»2«/mn»«/msub»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«msub»«mi»t«/mi»«mn»3«/mn»«/msub»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8230;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«msub»«mi»t«/mi»«mi»n«/mi»«/msub»«/mrow»«/mstyle»«/math», where «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 and «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» is the general term of the sequence.
Skill Builder
For more information on sequences, click the Skill Builder button to access the Skill Builder page.