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Sequences and Series

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Keith Poehleman

on 23 May 2014

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Transcript of Sequences and Series

Example:

Determining Convergence & Divergence with an Infinite Series
Determine whether the following series Converges or Diverges



Practice:
Objectives
Sequences and Series
The Integral Test
Convergence
and
Divergence

Infinite Series
Period 2
Mr. Eick
Chapter 9.1-9.3
.
.
Determining convergence or divergence of a series can usually be done
using several different tests. Today we will be going over Partial Sums.
Convergence and Divergence
Sequences and Series
in Calculus

By: Keith, Micela, Ysabela, Marlon
Objectives:
Definition:
If the sequence of partial sums approaches one specific number, the sequence
converges
to that number.
In certain problems, like the example below, you will see that series can converge to a specific number when the ratio is less than 1.
Convergence
Definition:
If the series does not approach one specific value, the series will approach infinity.
If the sequence approaches infinity, the sequence
diverges
.
In certain problems, like the example below, you will see the series also diverge when the ratio is greater than 1.
Divergence
Infinite Series
The Integral Test
If the function is positive, continuous, and decreasing, then both the function and series either converge or diverge.
Definition of a series:
Example:

The series converges because the ratio is less than 1, so the series converges to 6.
The series diverges because the ratio is greater than 1 and diverges to infinity.
Σ

n=0
3
----
2^n
By the end of this lesson, students should be able to:
- determine whether the infinite series converges or diverges
- understand the integral test and be able to apply it to find the divergence of a series.
To use the Integral Test:
1. To determine whether the function is decreasing, find the derivative.
2. Integrate the original function with respect to x.
3. Find the limit of the integrated function as x
approaches infinity.
Practice:

Example:
Example:
Sequences and Series
Definition of a Sequence:

Definition: a sequence of numbers in which an infinite number of terms are added successively in a given pattern.
A sequence is an ordered list of numbers. The numbers in this ordered list are called terms.
For example, the numbers "1, 2, 3, 4" are a sequence with the terms "1," "2," "3," and "4."
A series is the value you get when you add up all the terms of a sequence.
For example, if the sequence is "1, 2,
3, 4," the series will be the sum of
those values which is 10.
Partial Sums
n=1

Σ
n
-----------
(n^2)+1
Σ
n=1

1
-----
n+3
Σ a(n) or the nth partial sum=
S(n)= a(1) + a(2)..... +a(n) if said sequence converges. Then the series converges. However if the sequence diverges than so does the series.

n=1
Example
Σ

n=1
1/2(n)= 1/2+ 1/4+ 1/8+ 1/16.....
Partial Sums:
S(1)= 1/2
S(2)= 1/2 +1/4 = 3/4
S(3)= 1/2+ 1/4+ 1/8= 7/8
S(n)= 1/2+ 1/4+ 1/8+.....+ 1/2(n) =2^n-1/(2^n)
Because lim 2^n-1/(2^n)= 1, it converges.

Since there is a sum, it converges.
However, if lets say we are given the series



since it will eventually lead
to infinity the sum becomes ambiguous and therefore diverges.

n -> ∞
Practice
Determine whether {a(n)} and Σa(n) are convergent. When a(n) is equal to n+1/n

Σ
n=1
Remember that {a(n)} is the series not the sequence.
1= 1+1+1+1+1.....
Σ

n=1
Example:
Σ

n=0
(3/2)^n
Practice
Determine whether the following
series
Converges
or
Diverges
Σ

n=0
(7/6)^n
The ratio of the series is greater than 1, so the series diverges to infinity.
Bibliography:
"Pauls Online Notes : Calculus II - Integral Test." Pauls Online Notes : Calculus II - Integral Test. N.p., n.d. Web. 19 May 2014.
"Pauls Online Notes : Calculus II - Sequences." Pauls Online Notes : Calculus II - Sequences. N.p., n.d. Web. 20 May 2014.
"Pauls Online Notes : Calculus II - Series - The Basics." Pauls Online Notes : Calculus II - Series - The Basics. N.p., n.d. Web. 21 May 2014.
"Pauls Online Notes : Calculus II - Series & Sequences." Pauls Online Notes : Calculus II - Series & Sequences. N.p., n.d. Web. 18 May 2014.
"Pauls Online Notes : Calculus II - Series - Convergence/Divergence." Pauls Online Notes : Calculus II - Series - Convergence/Divergence. N.p., n.d. Web. 19 May 2014.
Section 8.3 The Integral And Comparison Tests, and 2010 Kiryl Tsishchanka. Section 8.3 The Integral and Comparison Tests (n.d.): n. pag. 2010. Web. 18 May 2014.
"Convergence of Series." Convergence of Series. N.p., n.d. Web. 22 May 2014.
Full transcript