Unit 1B

Limits

Lesson 6: Limits of Sequences and Series

Around 450 BC, a Greek philosopher named Zeno of Elea proposed a thought experiment that, at first glance, seemed to show that a horse running a race could never reach the finish line. Imagine a horse running down a track. Before it can travel the full «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»100«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/mrow»«/mstyle»«/math» distance, it must travel half the distance («math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»50«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/mstyle»«/math»). In order to travel the next «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»50«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/mstyle»«/math», it must first travel half of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»50«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/mstyle»«/math» («math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»25«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/mstyle»«/math»). And, in order to travel the next «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»25«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/mstyle»«/math», it must first travel half of that «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»25«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/mstyle»«/math» distance («math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»12«/mn»«mo».«/mo»«mn»5«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/mstyle»«/math»). Time is required for the horse to travel each half-distance interval, and there are an infinite number of half-distance intervals.

The video below, Zeno’s Race Horse Paradox, shows the distance travelled in the corresponding time interval. The video also provides some insight into the foundations of “The Calculus ”.

Zeno’s argument was that since it would require an infinite number of time intervals, and it is not possible for there to be an infinite number of intervals, the motion was not possible in the first place. Although this argument defies our everyday experiences, to mathematically disprove it requires the use of a series.

A series is the sum of the terms in a sequence and has the following form.


A series like the one in Zeno’s paradox is called an infinite geometric series. There is a constant ratio between each pair of consecutive terms.


where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»a«/mi»«/mstyle»«/math» is the first term and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»r«/mi»«mo»=«/mo»«mfrac»«msub»«mi»t«/mi»«mi»n«/mi»«/msub»«msub»«mi»t«/mi»«mrow»«mi»n«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/msub»«/mfrac»«/mrow»«/mstyle»«/math» is the common ratio of the series.

To resolve Zeno’s paradox, however, the sum an infinite number of terms must be found. As seen with the limits of sequences, there are many cases for which it is possible to determine the limit at infinity.

For an infinite series, the sum is given by the following formula.