Quick Instructions
Add Series and Sequences content to this page. See specific details on the home page. This unit has been split into parts A & B. Content between the two must be cohesive as it is one continuous, connected material. They are split simply to better organize the page.


Series and Sequences (Unit 12A) Learning Targets
  • Write and use explicit and recursive sequences
  • Use summation (sigma) notation.
  • To find sums of finite series and sums of convergent infinite geometric series.
  • To determine the interval of convergence of series in geometric form.
  • Use mathematical induction to prove conjectures.

Types of Rules


Explicit Rule = rule to evaluate the nth term of a sequence directly

Recursive Rule = rule to evaluate the nth term only by knowing the values of the previous (n-1)th and start terms


Sequences


Arithmetic

Arithmetic Sequence = a sequence of numbers in which the difference between consecutive terms is constant
Ex: 20, 24, 28, 32, 36 …

Equation to find the nth term in an arithmetic sequence:


‍‍Example:
Write an explicit rule for the sequence: 20, 24, 28, 32, 36...






Geometric
Geometric Sequence = a sequence of numbers in which the common ratio between consecutive terms is constant
Ex: 3, 9, 27, 81, 243 …

Equation to find the nth term in a geometric sequence:


Example:
Write an explicit rule for the sequence: 3, 6i, -12, -24i...






Summation


Sigma (Summation) Notation = a compact way of expressing the sum of a sequence of numbers
i.e. Evaluate f(n) for all integers starting at n = i and stopping at n = k, then summing the results



f(n): function
n = i : start
n = k: stop

Find Summation by Hand




n = 3 --> = -27
n = 4 ... = 64
n = 5 ... = -125
n = 6 ... = 216
n = 7 ... = -343

-27 + 64 + (-125) +216 +(-343) = -215

= -215

Find Summation with a Calculator

TI 83/84 Command: sum(seq(rule, variable, start, stop))
~sum: 2nd->STAT->MATH->5
~seq: 2nd->STAT->OPS->5
TI 89 Command: F3, 4 then Σ(rule, variable, start, stop)


Partial Sums and Infinite Sums


Partial Sums & Series = partial sums refer to the summation of the n amount of terms in a series

Arithmetic Partial Sum

Partial sum of any arithmetic sequence:

*also known as the nth partial sum (or finite sum) of an arithmetic sequence

Geometric Partial Sum

Partial sum of any geometric sequence:

*also known as the nth partial sum (or finite sum) of a geometric sequence

Arithmetic Infinite Sum

Infinite sum of an arithmetic sequence:
‍‍Does not exist! Logically, unlike a geometric sequence, arithmetic sequences are linear and not asymptotic. Therefore, you never reach a single number; it gets higher and higher.

Geometric Infinite Sum

‍‍Infinite sum of a geometric sequence:


Key Terms

‍Converge

When the partial sums of a geometric series approach a finite number; when |r|< 1
*An infinite geometric series converges when |r|<1
If |r|<1, then the sum of an infinite geometric series can be found.

Diverge

When the partial sums of a geometric series do NOT approach a finite number; when

*An infinite geometric series diverges when




Example - Geometric Series Infinite Sum:
Write an infinite geometric series, using sigma notation, whose result is

(Find a rule whose limit as n approaches infinity results in that equation)

--> Recall:
An infinite geometric series summation has the equation:

So you set that equal to the f(x) equation to fit the f(x) equation to the format of the infinite series formula (i.e. get the denominator to become 1 - r)

and you get a = 3/4 and r = -x/4
With this, you plug in the a value and r value in the geometric sequence formula.
--> Recall:
Equation to find the nth term in a geometric sequence:


But in PRECALC HONORS AND CALC BC, we don't like this. So we manipulate the algebra to isolate for the x value.

So now the x value is alone and that's the answer.
If asked to identify the interval over which the series converges, |r| < 1. Since r = -x/4

-4 < x < 4
The series converges over the interval (-4, 4).



Example 1 - Interval of Convergence:
Evaluate the interval over which the given value r of a geometric series converges.

|r| <1
Put

into r:

Change the equation to:

Multiply |x+3| both sides:
1<|x+3|
Separate into two equation:
x+3>1 or x+3<-1
Solve x
x> -2 and x<-4
Write the interval notation:




Example 2 - Interval of Convergence:
For what values of x will the following series converge? (Interval notation)

By looking at the second term, divide it by the first term to get the ratio

is the ratio (i.e. multiply that ratio by the first term and you get the second term, etc. etc.)
When you want to find where the series converges, |r|<1
so -1 < r < 1

You know the value of 2(x-1) has to be greater than 5 because the magnitude of the ratio has to be less than one.
2(x-1) > 5
2x > 7
x > 7/2
-5 > 2(x-1)
-3 > 2x
x < -3/2
So the interval at which the series converges is



Telescoping Sums


Telescoping Sum = a sum in which subsequent terms in a series cancel out, leaving only initial and final terms; used with decomposing partial fractions



Example - Telescoping Sums:

Given

Find

First, decompose the function into partial fractions using the "Cover Up" method

You get A = 1 and B = -1
So then,

‍‍the middle terms cancel out (1/2 would cancel out -1/2, etc.) so only‍‍

is left. Now you can substitute 100 for n to solve for the sum of the first 100 terms.



Proof by Induction

This is not a direct proof.

Goal: To apply "proof by mathematical induction" as a method to prove that a given series can be represented by a specific rule.

Step-by-Step Process:
1. Let’s show the rule is true for the base case; for n=1. (Prove that the first statement is true; usually check for k=1.)
2. If the result is true, then assume it will be true for n=k. Then, I will show that the rule also applies for n=k+1. (Now since the given statement is true for all n=k, we need to show that it is true for n=k+1.)
3. Write a concluding statement.


This is a really cool video that explains the basics of induction!
http://www.youtube.com/watch?v=IFqna5F0kW8



Example 1 - Proof by Induction:
→ Question: Prove by induction that the


→ Put the rule of this series inside the parentheses.
→ Has a common ratio of 2 so this is a geometry series.



1. Check base case. n=1.



2. Since base case true, we assume the rule is true for n=k and must show that it is also true for n=k+1.



3. By induction, we have shown the rule

is true for all values of n.



Example 2 - Proof by Induction:
→ Question: Prove by induction that

CodeCogsEqn.gif

is true.
Rewrite in form we like:
CodeCogsEqn_(1).gif

Follow the same steps 1-3 above as before.



Primary authors of this page (as of 06/02/12):

Jenny Li
Tina Moazezi


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