1. Calculus on the wall mastermathmentor.com presents Stu Schwartz
Helping students learn … and teachers teach
21. Infinite Series
Created by:
Stu Schwartz
Un-narrated Version
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2. Calculating Pi
We know that the circumference C of a circle is calculated by multiplying π times the diameter. So π is the ratio of the circumference to the diameter. But how do we calculate the value of π without this geometric approach?
Circumference
diameter

3. Vocabulary Terms used in this slide show: • Sequence • p-series test
• Recursive formula • Harmonic series
• Explicit formula • Alternating series test
• Convergent sequence • Telescoping series
• Divergent sequence • Integral test
• Series • Direct comparison test
• Infinite series • Limit comparison test
• Convergent series • Ratio test
• Divergent series • Absolutely convergent
• Convergence test • Conditionally convergent
• nth term test • Root test
• Geometric test

4. Sequences
We define a sequence as a set of numbers that has an identified first member, 2nd member, 3rd member, and so on. We use subscript notation to denote sequences: a1, a2, a3, etc. Sequences are rarely made up of random numbers; they usually have a pattern to them.
What is the 4th and 5th members of the sequence: 1, 2, 4, … ?
If you said that the 4th member is 8 and the 5th member is 16, the pattern is: to get the next term, we double the previous term.
If you said that the 4th member is 7 and the 5th member is 11, the pattern is: we add 1 to the first member, add 2 to the 2nd member and thus add 3 to the 3rd member.
The problem with giving the terms of a sequence is that there is an assumption that the student can see the pattern of the terms. Sometimes the pattern isn’t obvious, even if many terms are given. For instance, finding the next term in the sequence 1, 2, 3/2, 2/3, 5/24 … is quite difficult, and yet when you see the answer in the next slide, it makes perfect sense.

5. Defining Sequences
Rather than give the terms of a sequence, we give a formula for the nth term an. There are two ways to do so: recursively and explictly.
Recursive: we give the first term a1 and the formula for the nth term an in terms of a function of the (n - 1)st term, an-1 .
Explicit: we give a formula for the nth term an in terms of n.
Recursive: a1 = 1, an = 2an-1
The 5th term would be 16.
Advantage: pattern easy to see.
Disadvantage: to find the 10th term, we need the 9th which needs the 8th, …
For the sequence: 1, 2, 4, 8, …
Explicit: an = (1/6)n3 - (1/2)n2 + (4/3)
The 5th term would be 15.
Advantage: easy to find the 10th term.
Disadvantage: formula is not obvious.
In this course, students rarely will be asked to generate a formula for the nth term from the terms themselves. However, if they are given the nth term formula, which is usually given explicitly, they should be able to find any term. For instance, the sequence 1, 2, 3/2, 2/3, 5/24, … is defined as an = n2/n!. The 6th term is 36/720 = 1/20.

6. Convergent and Divergent Sequences
Sequences either converge (are convergent) or diverge (are divergent).
Convergent sequences: the limit of the nth term as n approaches ∞ exists
Divergent sequences: the limit of the nth term as n approaches ∞ does not exist

7. Using L’Hospital’s Rule
Whenever you are asked about the convergence or divergence of a sequence when given an, it is best to write out a few terms of the sequence to get a sense of it. Even then, you can get fooled. To be sure, you can use L’Hospital’s rule to find the limit of an as n approaches infinity, and if L’Hospital’s rule cannot be used, use simple logic.

8. Series
We define a series as the sum of the members of a sequence starting at the first term and ending at the nth term:
We define an infinite series as the sum of the members of a sequence starting at the first term and never ending.
The remainder of this slide show is centered whether an infinite series converges (has a limit). A sequence converges if its nth term an has a limit as n approaches infinity. A series converges if the sum of its terms has a limit as n approaches infinity. Determine whether each of the following series converges.

9. Convergence Tests
A convergence test is a procedure to determine whether an infinite series given the formula for an is convergent. Some convergence tests can be done by inspection while others involve a bit of work that will need to be shown.
The nth term test
If the nth term does not converge to zero, the series must diverge.
If the nth term does converge to zero, the series can converge.
the sequence terms
the partial series

10. Geometric Series & Convergence
A series in the form of is a geometric series.
The Geometric test
If |r| ≥ 1, the series diverges.
If |r| < 1, the series converges to a/(1 – r)
The figure is a square of side 8 and the midpoints of the square are vertices of an inscribed square with the pattern continuing forever. Show that the sum of the areas and perimeters are convergent.

11. The p-Series and Harmonic Series
A series in the form of where p > 0, is a p-series.
The p-series test:
If 0 < p ≤ 1, the series diverges.
If p > 1, the series converges.
p is a positive constant:
the sequence terms
the partial series
Any series in the form of
∑ [(c/an + b)] is called an harmonic series and is divergent.

12. More on the Harmonic Series 1/n
It seems counter-intuitive that is divergent. Here’s a simple proof:
It takes 31 books for the overhang to be 2 books long, 227 books for the overhang to be to be 3 books long, 1,674 books for the tower to be 4 books long, and over 272 million books for the overhang to be 10 books long. Try it with a deck of cards.
Deck of
52 cards
The length of the overhang is where n is the number of books. This is the harmonic series and thus theoretically, it will balance with an infinite number of books. ⅙ …
Suppose we stack identical books of length 1 so that the top book overhangs the book below it by ½, which overhangs the book below it by ¼, which overhangs the book below it by ⅙, etc. This structure, called the Leaning Tower of Lire will (just barely) balance. ⅛
1/10

13. Alternating Series
An alternating series is one whose terms alternate in signs.
The Alternating series test:
Error in an alternating series
In a convergent alternating series, the error in approximating the value of the series using N terms is the (N + 1)st term .

14. Telescoping Series
However, it is possible that we subtract one divergent series from another, our answer converges.
A telescoping series is an alternating series in the form of If this passes the nth term test, the series converges. Expand the expression.
If we take a divergent series and subtract a divergent series, we can get another divergent series.

15. The Integral Test
If the integral test shows convergence of a series, the value of the integral is not the value of the series. It is merely an indicator that the series converges.

16. Comparison Tests
If there is a deviation from the forms already studied, the tests cannot be used.
The Direct Comparison test:

17. The Limit Comparison Test

18. The Ratio Test
The Ratio test:

19. Failure of the Ratio Test
You may wonder: if the ratio test is so versatile, why do we need all of the other tests? Why not just apply the ratio test immediately?
The ratio test will always be inconclusive with a series in the form of a polynomial over a polynomial or polynomials under radicals.
∑an absolutely convergent:
∑an converges and ∑|an | converges
∑an conditionally convergent:
∑an converges and ∑|an | diverges

20. The Root Test The Root test:
The following numbers are written from smallest to largest:
n20, 20n, n!, nn.
While an expression of a smaller expression over a larger expression could converge, an expression of a larger expression over a smaller expression diverges.
The Root test:

21. Series Convergence/Divergence Flowchart
nth term test No
Yes
Geometric test No No
p-series test
Yes No
Alternating series test
Yes
Is the series telescoping?

22. More Tests in Order of Usefulness
Ratio test
Yes No
Limit Comparison test
Yes No
Root test
Yes No
Integral test
Yes No

23. Vocabulary Do you understand each term? • Sequence • p-series test
• Recursive formula • Harmonic series
• Explicit formula • Alternating series test
• Convergent sequence • Telescoping series
• Divergent sequence • Integral test
• Series • Direct comparison test
• Infinite series • Limit comparison test
• Convergent series • Ratio test
• Divergent series • Absolutely convergent
• Convergence test • Conditionally convergent
• nth term test • Root test
• Geometric test