Convergence of Sequences and Series
(by Mike May, S.J.)

MT A14303                                                                                                             Fall 2001

Notes on Chapter 11 (Sequences and Series)

Background idea  Chapter 11 takes the idea of Taylor polynomials, that we approximate a function f(x) at the point x = a with the nth  degree Taylor polynomial,

,

and extends n to infinity.  Unfortunately, there is a lot of technical details to work through on the way, so students often get lots before they get to the punch line.  Thus it seems worthwhile to start with a day of overview.

Visualizing the main idea  The simplest way to visualize the limit series

is to construct an animation of Taylor polynomials with increasing values of n.  We do that with a Maple worksheet, TaylorAnimation.mws.

Preparing for the rest of the chapter  Seeing the goal lets us comment on some of the details that we waived our hands about for the Taylor series.  A Taylor series is defined at each point as a sum of infinitely many terms.  We only know how to add together finitely many things.  Hence to understand our limit of approximating polynomials we have to ask when it makes sense to add up infinitely many things.  That leads us into the bulk of the chapter on sequences and series.  We view the infinite sum as the limit of the finite sums as the range goes to infinity.  It makes it useful to look at an applet to visualize finite sums.  (This can actually be done on your calculator.  The computer is simply much faster.)

Open Internet explorer to the URL:

http://www.slu.edu/classes/maymk/MathApplets-SLU.html

Select sequences and series.  The applet graphs the series starting with n=nStart.  (If the general term for the series has an n in a denominator, we want to start the series at 1.  Other times we want to view the series with the domain of n starting at 0.)  It shows a mark for each nStep.  It should show about 100 points.  You can put is 3 arbitrary series and change the parameters of three standard series, a geometric, a harmonic, and an exponential series.  You can view terms, sums, or ratios of terms.  You can also evaluate for a specified n (nView).

1) Set “n to view” to 110 to find the 110th term of the 5 default series.

2) Set “n to view” to 200 to find value of the sum of the first 200 terms of the 5 default series.

3) For each of the 5 default series, make your best guess if the 5 series converge and if so to what.

For the convergent series, give an n when the sum settles down enough that the mark for the sum stops changing.

4) Explore the geometric series.  Turn off all the series but G(n).  Turn on both the terms and sum.

Change the starting point, n start to 0.  The series has an initial value (first box) and a ratio (second box).  Set the initial value to 1.  Using the applet to explore, estimate the value the series converges to when the ration is .5, .75, .8, .9, and .99.  (You will have to adjust the range of y values shown for some of these ratios.)  Repeat the process when the ratio is -.5, -.75, -.8, and -.9.  What happens if the ration is 1 or -1?  What happens if the ratio is bigger than 1?

5) Explore the harmonic series (H(n)) and the alternating harmonic series (the default for B(n).)  Change n start to 1.  Turn off the series except H(n) and B(n).  To what value does the alternating harmonic series seem to converge.  Check if the series converges, only later by checking n step to 10, 100, and 1000.  How much does the series increase from 100, 000 to 300,000?

6) Look at Taylor series of sin(x).  The absolute value of the nth term of the sequence is

x^(2*n+1)/fact(2*n+1).  Compare this sequence to the geometric sequence with initial value 1 and ratio .  If x=10, when is the size of the size series dominated by the geometric series?  When are the terms of the series dominated by the size of terms of the geometric series?  Repeat the question when x=25.