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EXAMPLES
Sequences	Geometric Series Plus	Alternating Series	Power Series	Taylor Series
Integral Test	Direct Comparison Test	Limit Comparison Test	Ratio Test	Root Test

If it is not already on your hard drive, you will need to download the free DPGraph Viewer to view some of the pictures linked to on this page.

QuickTime free download.

	Example for Exam III
	Take-Home (Team) Portion of Exam III
	Convergence Tests for Infinite Series from Harvey Mudd College
	At this site you can find some java tools that include a sequence grapher and a series grapher. Click on java tools at the bottom of the table of contents on the left.
	Hotmath You can look at free solutions to problems in exercise sets from a wide variety of mathematics textbooks including Calculus by Larson, Hostetler, Edwards, eighth edition. They now have chapters 1 - 12 available. Only a few solutions are still free.
	WIMS A really nice set of graphing tools by XIAO Gang which will produce pictures and graphs that can be saved. Included is a free online sequence and series grapher. Click on Sequence plot at the bottom of the page. Click on Sigma to compute finite and infinite series sums. Traces Animes at the same site is a terrific 2D and 3D plotter.
	Maple Worksheet on computing sequences and sums
	Maple Worksheet on sequences and series with graphs
	Maple Worksheet with shorter commands for graphing sequences and series
	Maple Worksheet graphing a Power Series and constructing and graphing a Taylor polynomial
	Maple Worksheet with tan(x), exp(x), ln(x)--constructing and graphing Taylor and Maclaurin polynomials
	Maple Worksheet on graphing functions defined by power series using finite series approximations
	YouTube proof that the sum of all natural numbers is -1/12

9.1 You need to be able to determine whether or not a sequence converges and if it does converge be able to find its limit. You need to be able to recognize some common sequence patterns and given the first few terms in the sequence be able to write an expression for the nth term of the sequence that at least works for the given first few terms.

To the right is a plot of the sequence a_n = (8n³ - 4n²)/(n³ + 20n + 10) as n goes from 1 to 50. What would the corresponding infinite sequence converge to?

9.2 You need to distinguish between the term "sequence" and "series". You need to understand and be able to use summation notation. You need to understand the terminology and definition of convergent and divergent series given on page 606. You need to be able to recognize a telescoping series, determine whether or not it converges, and compute what it converges to if it does converge. You need to be able to recognize a geometric series and be able to sum (determine what it converges to) a convergent infinite geometric series. You should also be able to sum any finite geometric series. You need to know and be able to apply theorems 9.8 and 9.9. Be able to prove that the sphereflake , P615:108, has an infinite surface area. Here are some drill problems on geometric series, the nth term test, and telescoping series. You should be able to compute the total distance traveled by an idealized bouncing ball dropped from a given initial height that always bounces back up to some fraction (less than one) of the height it just fell from. The first five people to correctly identify the fraction (ratio r) in this bouncing ball animation will receive a bonus.

a_n = 10(3/4)ⁿ

The picture on the left shows a graph of the first 51 terms of a geometric sequence and the picture on the right shows a graph of the sequence of partial sums of the corresponding geometricseries. What does the geometric sequence converge to? What does the infinite geometric series converge to?

S_n = 40(1 - (3/4)ⁿ)

9.3 You need to be able to apply the Integral Test to appropriate infinite series to prove that the series either converges or diverges. We will also discuss how to extend the Integral Test in order to put a bound on the error in summing a finite number of terms in an attempt to approximate the value a convergent infinite series converges to. Here are somedrill problems dealing with the Integral Test. You need to know what a p-series is, what a harmonic series is, and how to apply Theorem 9.11.

9.4 You need to be able to apply the Direct Comparison Test and the Limit Comparison Test to determine whether certain types of infinite series converge or diverge. Here are some drill problems on the Limit Comparison Test.

9.5 You need to know what an alternating series is. You need to be able to apply the Alternating Series Test to determine convergence of series to which the test applies and to be able to put a bound on the error in summing the first n terms of a convergent alternating series by using the Alternating Series Remainder Theorem (P 633). You need to be able to approximate the sum of a convergent alternating series to a specified level of accuracy by using your calculator or computer. Here are some drill problems dealing with alternating series. You need to be able to determine whether a convergent series converges absolutely or conditionally.

9.6 You need to be able to apply the Ratio Test and the Root Test to test for absolute convergence. You will not have to use the Root Test since the Ration Test will handle all the examples we do in class. Here are some drill problems on the Ratio Test. Here are more drill problems focusing on trying to determine which test to use when testing for convergence.

Here is the HMC summary of convergence tests for infinite series. Here is a link that includes some convergence tests and some common series expansions.

9.7 You need to be able to construct Taylor and Maclaurin polynomials to approximate given functions. You need to be able to apply Taylor's Theorem (P 654) to put a bound on the error in using a Taylor Polynomial to approximate the value of a given function for a given value of the independent variable (usually x). Here is an applet that will graphTaylor approximations to the sine function and another one that will graph Taylor approximations to exp(x). Here is a nice Quicktime animation that looks at Maclaurin polynomials approximating f(x) = sin(x) + cos(x). Below are the first, third, fifth, seventh, ninth, eleventh, and thirteenth Maclaurin polynomials approximating the sine function (sine function in blue).

Here is a short PowerPoint presentation showing the 1st through the 23rd Maclaurin polynomials approximating the sine function and a slightly longer version.

9.8 You need to be able to recognize the type of infinite series that would be called a power series centered at c. You need to be able to find the domain of a function defined by a power series. This means you will need to be able to find the radius of convergence and interval of convergence of a power series including being able to check for endpoint convergence. You need to know the conditions under which you can differentiate and integrate power series and be able to do it.

9.9 You will need to be able to use the formula for summing a convergent infinite geometric series to construct the power series representation for a variety of functions. You will need to be able to use "Operations with Power Series", page 671, to construct power series representations for functions from the power series representations of other functions. You will need to be able to apply the technique of finding a power series for a given function by integrating the power series for another function. To the right is the graph of y = ln(x) in blue and the graph of the Taylor polynomial P₆(x) centered at 1 approximating ln(x) in red.

P₆(x) = (x-1) - (1/2)(x-1)² + (1/3)(x-1)³ - (1/4)(x-1)⁴ + (1/5)(x-1)⁵ - (1/6)(x-1)⁶

What would be the interval of convergence of the corresponding infinite series?

9.10 You need to know the definition of Taylor and Maclaurin series. You need to know the meaning of Theorem 9.23, Convergence of Taylor Series. In particular you need to understand R_n(x) (some call this R_n+1(x)) in Taylor's Theorem. You need to be familiar with the power series representations for elementary functions shown on page 682 of your text and be able to use these results to construct power series representations of other functions. You need to be able to use power series to approximate definite integrals that you would otherwise not be able to evaluate using the Fundamental Theorem of Calculus. At the right is the graph of y = e^x in blue and the Maclaurin polynomial P₆(x) approximating e^x in red.

You need to be able to compare on your graphing calculator or computer the graphs of a given function, say sin(x), and its nth Taylor or Maclaurin polynomial, P_n(x), and be able to determine graphically an interval over which the function values of sin(x) and P_n(x) differ by less than a prescribed amount, say 0.00001.

Zeno's Paradox

The Greek philosopher Zeno (495 - 435 B.C.) may have been a fast runner at one time and perhaps claimed that if he was given a slight head start no one on earth could catch him. If so then by the paradox named after him he would still have made the claim even after he aged and was no longer a fast runner. This is because, as he said, to catch him one must get to where he was and by then he will have moved forward. You can never get to where he is because to get to where he is you must get to where he was first and by the time you get to where he was he will no longer be there. This implies that to score a touchdown in American football a wide receiver has merely to catch the ball behind the secondary. No matter how fast the defensive backs are they will not be able to catch up to the receiver and the receiver can simply walk to the end zone, taking care never to pause along the way. As long as he never stops moving and continues heading toward the end zone no one can ever reach his position from behind him. He would have to stop moving away from the quarterback briefly to catch the football since otherwise the football could never catch up to him.

To present this paradox mathematically, Zeno considered the case of Achilles and the tortoise. I will consider an aged Achilles and the tortoise although the same argument would apply to a young fleet Achilles and the tortoise. Suppose Achilles runs at a rate of 1 meter/sec and the tortoise crawls at a rate of 9/10 meter/sec. If the tortoise is given a 1 meter head start and continues to travel at a rate of 9/10 m/sec then Achilles traveling at 1 m/sec cannot catch up to him. This is because once Achilles has traveled the 1 meter to get to where the tortoise was when the chase began, the tortoise will have traveled 9/10 of a meter farther. Once Achilles travels that 9/10 of a meter the tortoise will have traveled an additional (9/10)² meter or 81/100 meter. Once Achilles travels this 81/100 meter the tortoise will have traveled an additional (9/10)³ meter or 729/1000 meter, and so on so Achilles will never catch the tortoise. Achilles will travel a distance of

and in spite of n approaching infinity he will never catch the tortoise. Can you mathematically demonstrate the flaw in this argument?

In the animation of this paradox the tortoise is in blue, Achilles in red, and the time that has elapsed during the chase is shown in green along the y-axis. Quicktime animation