CHAPTER 9 INFINITE SERIES THEOREMS BOUNDED MONOTONIC SEQUENCE

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Bounded Monotonic Sequence

Chapter 9: Infinite Series - Theorems



Bounded Monotonic Sequence

If a sequence {an} is bounded and monotonic, then it converges.


Convergence of a Geometric Sequence

A geometric series with ratio r diverges if CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE . If CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE , then the series converges to the sum

CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE , CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE


nth Term

If CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE converges, then CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE .

If CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE , then CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE diverges.


Integral Test

If f is positive, continuous, and decreasing for all CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE and CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE , then

CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE and CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE

either both converge or both diverge. (Note: These conditions need only be satisfied for all CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE .)


p-Series

The p-series

CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE

  1. converges if CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE , and

  2. diverges if CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE .


Direct Comparison Test

Let CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE for all n.

  1. If CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE converges, then CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE converges.

  2. If CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE diverges, then CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE diverges.


Limit Comparison Test

Suppose that CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE , CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE , and

CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE

where L is finite and positive. Then the two series CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE and CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE either both converge or both diverge.


Alternating Series Test

Let CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE . The alternating series

CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE and CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE

converge if the following two conditions are met.

  1. CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE

  2. CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE , for all n*


* This can be modified to require only that CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE for all n greater than some integer N.


Alternating Series Remainder

CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE


Absolute Convergence

If the series CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE converges, then the series CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE also converges.


Ratio Test

Let CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE be a series of non-zero terms.

  1. CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE converges absolutely if CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE .

  2. CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE diverges if CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE or CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE .

  3. The Ratio Test is inconclusive if CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE .


Root Test

Let CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE be a series.


  1. CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE converges absolutely if CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE .

  2. CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE diverges if CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE or CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE .

  3. The Root Test is inconclusive if CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE .

Taylor Polynomial

If f has n derivatives at c, then the polynomial

CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE

is called the nth Taylor polynomial for f at c. If CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE , then

CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE

is also called the nth Maclaurin polynomial for f.


Power Series

If x is a variable, then an infinite series of the form

CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE

is called a power series. More generally, and infinite series of the form

CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE

is called a power series centered at c, where c is a constant.


Convergence of a Power Series

For a power series centered at c, precisely one of the following is true.


  1. The series converges only at c.

  2. There exists a real number CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE such that the series converges absolutely for CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE , and diverges for CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE .

  3. The series converges absolutely for all x.


The number R is the radius of convergence of the power series. If the series converges only at c, CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE , and if the series converges for all x, CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE . The set of values of x for which the power series converges is the interval of convergence of the power series.


The Form of a Convergent Power Series

If f is represented by a power series CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE for all x in an open interval I containing c, then CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE and

CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE


Taylor Series

If a function f has derivatives of all orders at CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE , then the series

CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE

is called the Taylor series for CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE at c. Moreover, if CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE , then the series is the Maclaurin series for f.


Guideline for Finding a Taylor Series

  1. Differentiate CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE several times and evaluate each derivative at c.

CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE

Try to recognize a pattern in these numbers.


  1. Use the sequence developed in the first step to form the Taylor coefficients CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE , and determine the interval of convergence for the resulting power series

CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE

  1. Within this interval of convergence, determine whether or not the series converges to CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE .



Power Series for Elementary Functions

Function


Interval of Convergence

CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE


CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE

CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE


CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE

CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE


CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE

CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE


CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE

CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE


CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE

CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE


CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE

CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE


CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE

CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE


CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE

CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE


CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE *

* The convergence at CHAPTER 9 INFINITE SERIES  THEOREMS BOUNDED MONOTONIC SEQUENCE depends on the value of k.




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