MATH 257 Calculus III								Week of April 7, 2003
Lab 2: Serious Series
Name 1
Name 2
Section:

I.  Laboratory Objectives
	to get down with series, geometric series, 
	and the integral test

II.  Formatting and Syntax Information:
	Use the Basic Input Palette to find  
	(∑_(n = 1)^∞) (Subscript[a, n])    and   (∫_a^b) f[x] dx

Infinite Series

An infinite series is an infinite sum.  
In the infinite sequence, {(Subscript[a, n])} represents an infinite list of values:
	{(Subscript[a, 1]),  (Subscript[a, 2]),  (Subscript[a, 3]),  . . . }
In an infinite series, (∑_(n = 1)^∞) (Subscript[a, n]) represents an infinite sum:
	(Subscript[a, 1]) + (Subscript[a, 2]) + (Subscript[a, 3]) + . . .
	
Summations can be finite or infinite.  Let's start with a finite sum.  The sum below has ten terms being added.  Execute the statement below:

(∑_(n = 1)^10)(( 5 - n)/(3  (n^5)))

To find the decimal approximation of this fraction to 20 significant digits, use the Mathematica command below.  The syntax is N for numeric value, the % sign refers to the previous computation and 20 is the number of significant digits you desire when computing the numeric value.

N[%, 20]

What you have now calculated is the 10th partial sum or (Subscript[S, 10])of the infinite series (∑_(n = 1)^∞)(( 5 - n)/(3  (n^5))).  Suppose you wanted the sum of the first 150 terms of this infinite series:

(∑_(n = 1)^150)(( 5 - n)/(3  (n^5)))

This huge fraction is hard to read and even harder to estimate, so let's once again ask Mathematica for a decimal approximation:

N[%, 20]

It seems that our series is converging. Mathematica can check this for us.

(∑_(n = 1)^∞)(( 5 - n)/(3  (n^5)))

This is not much help unless you are familiar with the Zeta function.  Let's ask for a decimal approximation:

N[%, 20]

Another way to see what is going on would be to make a ListPlot of the sequence of partial sums S[n].  The following will do this:

S[
        n_] := (∑_(k = 1)^n)(( 5 - k)/(3  (k^5)))

See how we had to use k instead of n there?  That's because we wanted the sum to go from 1 to n, so we needed a second variable to keep track of this.  Now we can ListPlot it using the Table command.  Let's plot it for n=1 to 100:

ListPlot[
      Table[{n, S[n]}, {n, 1, 100}]]

Geometric Series

Geometric Series are a specific type of infinite series.  Geometric Series have the form:

	a +  ar  +  \(TraditionalForm\`ar\^2\  + \ ar\^3\  + \ 
          ar\^\(\(4\)\(\ \)\) + \ \(\(.\)\(\ \)\(.\)\(\ \)\(.\)\)\ \  = \ ∑\+\(n = 1\)\%∞ ar\^\(n - 1\)\)

	a is the first term of the series and can be any real number
	r is the ratio of adjacent terms.   It can also be any real number.

When | r | < 1, the Geometric Series converges, but
when | r | ≥ 1, the Geometric Series diverges.
The sum of a convergent Geometric Series is (a/(1 - r))

Let's look at a few examples:
The series         (∑_(n = 1)^∞)4 ((((3/5)))^(n - 1)) 
\(has\ a\  = \ \(4\ and\ r\  = \ 
          3/5. \ Since\  | r\  | \ \(\(<\)\(\ \)\(1\)\)\), \ 
      this\ series\ will\ converge . 
Let' 
        s\ look\ at\ the\ nth\ partial\ sums . \ \ First, \ 
      define\ the\ sequence\ of\ partial\ sums, \ 
      and\ then\ evaluate\ \(\(it\)\(:\)\)\)

S[n_]:= (∑_(k = 1)^n)4 ((((3/5)))^(k - 1))

S[1]

S[2]

N[%]

S[5]

N[%]

S[10]

N[%]

Try getting this next one as a decimal to 20 significant digits:

S[50]

S[100]

N[%, 30]

S[n]

To find the limit of the nth partial sum:

Limit[S[n],  
      n ->  ∞]

Of course, we could have asked Mathematica to compute this infinite sum directly:

(∑_(n = 1)^∞)4 ((((3/5)))^(n - 1))

The Integral Test on Mathematica

Many infinite series are not geometric, and so a large portion of this course is spent developing tests for infinite series which tell us if the series converges or diverges. When geometric series converge, we can always find the sum of the series: S = (a/(1 - r)).  For most series, however, we will only be able to tell if the series converges or diverges . .  finding the sum of the series is often just too difficult by hand.

Integral Test:
If f(x) is a continuous, positive and decreasing function on [1, ∞) and 
(Subscript[a, n])= f(n) for each term in the positive termed series (∑_(n = 1)^∞) (Subscript[a, n]),
then (∑_(n = 1)^∞) (Subscript[a, n]) converges if and only if \(TraditionalForm
      \`\(∫\_1\%∞\( f(x)\) dx\ \)\)converges.

Look at the example (∑_(n = 1)^∞) (1/(n^2)).  
First check that the corresponding function f(x) = (1/(x^2))is continuous, positive and decreasing on [1,∞) by graphing f(x).

Plot[(1/(x^2)),  {x, 0, 20}]

Then check if the integral from 1 to ∞ converges or not:

(∫_1^∞) 
            (1/(x^2)) \[DifferentialD]x

Since the improper integral (∫_1^∞) 
              (1/(x^2)) dx is convergent, the infinite series 
(∑_(n = 1)^∞) (1/(n^2))is also convergent.

Mathematica can actually compute the sum of the series:

(∑_(n = 1)^∞) (1/(n^2))

Problems 
This is the part of the lab you will hand in for the lab.  Delete all the above stuff (except the title, your name, and section) before you submit your lab in Blackboard.

1.  Given the series  (∑_(n = 1)^∞) 6 ((5^(-n))):

a.)  Find the 5th partial sum, (Subscript[S, 5]) and its decimal approximation.

b.)  Find the 10th partial sum, (Subscript[S, 10]) and its decimal approximation to 10 significant digits.

c. )  Find the 20th partial sum, (Subscript[S, 20]) and its decimal approximation to 20 significant digits

d.)  What is the sum of the infinite series ?

e.)  Identify a and r for this series and compare the answer Mathematica just gave you with the computation of (a/(1  -  r)).

2.  Given the series (((∑_(n = 1)^∞) ((4/3)))^n)

a. )  Identify the first term a and the ratio r.

b.)  Do you think the infinite series converges or diverges?  Why?

c.)  Find the 20th partial sum (Subscript[S, 20]) and its decimal approximation to 15 significant digits .

d.)  Find the 100th partial sum(Subscript[S, 100]) and its decimal approximation to 15 significant digits.

3. Given the following series:   1 - (2/3) + (4/(9 ))- (8/27) + . . .

a. Write this infinite series using summation notation.

b.  Find the 10th partial sum, (Subscript[S, 10]) and its decimal approximation.

c.  Find the 20th partial sum (Subscript[S, 20 ])and its decimal approximation to 20 significant digits.

d.  At this point, what would you guess as the sum of the infinite series?

e.  Make a ListPlot of the partial sums (Subscript[S, n]) from n=1 to 50.

f.  Since this is a geometric series, identify a and r.  Use the formula for the sum of a convergent geometric series to find the sum of this series.

4.  Given the series (∑_(n=2)^∞)((ne)^(-n))

a.  Use the Plot command to show that f(x) = ((xe)^(-x))is a continuous, decreasing and positive function on the interval [2, 100].

b.  Now use Integral Test to show whether or not the series converges.

c.  Does the series  (∑_(n=2)^∞)((ne)^(-n))converge or diverge by the integral test?

d.  Compute the sum of the infinite series directly using Mathematica.

5.  Here's one that Mathematica can't handle very well: (∑_(n = 2)^∞) (1/(((ln(n)))^n))

a.  Use the Plot command to show that f(x) = (1/(((ln(x)))^x))is a continuous, decreasing and positive function on the interval [2, 100].  
     
	Remember , ln(x) is input as Log[x].

b.  Now use Integral Test to show whether or not the series converges.

c.  Does the series (∑_(n = 2)^∞) (1/(((ln(n)))^n))converge or diverge by the integral test?

d.  If you try to compute this series directly using Mathematica, it won't work.  So instead, define a sequence of the partial sums and use ListPlot[Table[ ]] to figure out what this series converges to.

BONUS: Sometimes a series will kick Mathematica's ass.  The following is one such series:  						(∑_(n = 2)^∞) Sin[
                n]/Log[n!].  
It is VERY hard to get Mathematica to tell you anything about this series.  But trust me: it does converge.  Do whatever you can to try to compute what it converges to with 6 decimals of accuracy.
WARNING!!!  Nasty series like this can cause Mathematica to blow up.  Make sure you sure you save your work before trying this one.

Now that you're done, 
	(1) 	Clean up your work by deleting everything that's not needed.  
		There should only be the title, your names, section number, and the answers to your questions
		(with any needed explanations, graphs, etc.).
	(2) 	Save this to your disk.
	(3)	Go back to Blackboard and upload your final lab report to the Digital Drop Box, located
		in the User Tools area.  Remember that you have to first ADD your file to your digital drop box and 
		then SEND it to me!