how to represent certain types of functions as sums of power series

Slides:

Advertisements

Similar presentations

9.2 day 2 Finding Common Maclaurin Series Liberty Bell, Philadelphia, PA.

Advertisements

Chapter 10 Infinite Series by: Anna Levina edited: Rhett Chien.

What is the sum of the following infinite series 1+x+x2+x3+…xn… where 0

Taylor’s Theorem Section 9.3a. While it is beautiful that certain functions can be represented exactly by infinite Taylor series, it is the inexact Taylor.

Section 9.2a. Do Now – Exploration 1 on p.469 Construct a polynomial with the following behavior at : Since, the constant coefficient is Since, the coefficient.

INFINITE SEQUENCES AND SERIES

Series (i.e., Sums) (3/22/06) As we have seen in many examples, the definite integral represents summing infinitely many quantities which are each infinitely.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 1.

Section 11.4 – Representing Functions with Power Series 10.5.

Calculus and Analytic Geometry II Cloud County Community College Spring, 2011 Instructor: Timothy L. Warkentin.

9.2 Taylor Series Quick Review Find a formula for the nth derivative of the function.

Chapter 1 Infinite Series. Definition of the Limit of a Sequence.

Taylor’s Polynomials & LaGrange Error Review

9.7 and 9.10 Taylor Polynomials and Taylor Series.

Sequences Lesson 8.1. Definition A __________________ of numbers Listed according to a given ___________________ Typically written as a 1, a 2, … a n.

TAYLOR AND MACLAURIN  how to represent certain types of functions as sums of power series  You might wonder why we would ever want to express a known.

MATH 6B CALCULUS II 11.3 Taylor Series. Determining the Coefficients of the Power Series Let We will determine the coefficient c k by taking derivatives.

In this section we develop general methods for finding power series representations. Suppose that f (x) is represented by a power series centered at.

Sect. 9-B LAGRANGE error or Remainder

9.1 Power Series.

This is an example of an infinite series. 1 1 Start with a square one unit by one unit: This series converges (approaches a limiting value.) Many series.

Remainder Theorem. The n-th Talor polynomial The polynomial is called the n-th Taylor polynomial for f about c.

Section 8.6/8.7: Taylor and Maclaurin Series Practice HW from Stewart Textbook (not to hand in) p. 604 # 3-15 odd, odd p. 615 # 5-25 odd, odd.

Geometric Series. In a geometric sequence, the ratio between consecutive terms is constant. The ratio is called the common ratio. Ex. 5, 15, 45, 135,...

9.3 Taylor’s Theorem: Error Analysis yes no.

Taylor series are used to estimate the value of functions (at least theoretically - now days we can usually use the calculator or computer to calculate.

Chapter 9 Infinite Series. 9.1 Sequences Warm Up: Find the next 3 terms… 1. 2, 6, 10, 14, … Common Diff: 4 18, 22, , 6, 12, 24, … Doubled Sequence.

9.1 Power Series Quick Review What you’ll learn about Geometric Series Representing Functions by Series Differentiation and Integration Identifying.

Series A series is the sum of the terms of a sequence.

Math 20-1 Chapter 1 Sequences and Series 1.5 Infinite Geometric Series Teacher Notes.

Representations of Functions as Power Series INFINITE SEQUENCES AND SERIES In this section, we will learn: How to represent certain functions as sums of.

Taylor and MacLaurin Series Lesson 8.8. Taylor & Maclaurin Polynomials Consider a function f(x) that can be differentiated n times on some interval I.

Geometric Sequences and Series Notes 9.2. Notes 9.2 Geometric Sequences  a n =a 1 r n-1 a 1 is the first term r is the ratio n is the number of terms.

Power Series Section 9.1. If the infinite series converges, then.

PARAMETRIC EQUATIONS Sketch Translate to Translate from Finds rates of change i.e. Find slopes of tangent lines Find equations of tangent lines Horizontal.

Warm Up Determine the interval of convergence for the series:

Chapter 7 Infinite Series. 7.6 Expand a function into the sine series and cosine series 3 The Fourier series of a function of period 2l 1 The Fourier.

Power Series A power series is an infinite polynomial.

Final Exam Term121Term112 Improper Integral and Ch10 16 Others 12 Term121Term112 Others (Techniques of Integrations) 88 Others-Others 44 Remark: ( 24 )

In the special case c = 0, T (x) is also called the Maclaurin Series: THEOREM 1 Taylor Series Expansion If f (x) is represented by a power series.

Section 11.3 – Power Series.

Infinite Sequences and Series

PROGRAMME 12 SERIES 2.

Finding Common Maclaurin Series

For the geometric series below, what is the limit as n →∞ of the ratio of the n + 1 term to the n term?

Section 11.4 – Representing Functions with Power Series

Section 11.3 Power Series.

3 TESTS Sec 11.3: THE INTEGRAL TEST Sec 11.4: THE COMPARISON TESTS

Find the sums of these geometric series:

Find the Taylor series for f (x) centered at a = 8. {image} .

Taylor Series and Maclaurin Series

Representation of Functions by Power Series (9.9)

Applications of Taylor Series

Sum and Difference Identities

MATH 1330 Section 6.1.

Section 2: Intro to Taylor Polynomials

how to represent certain types of functions as a power series

Infinite Sequences and Series

9.2 day 2 Maclaurin Series Liberty Bell, Philadelphia, PA

Math 20-1 Chapter 1 Sequences and Series

Finding Common Maclaurin Series

MACLAURIN SERIES how to represent certain types of functions as sums of power series You might wonder why we would ever want to express a known function.

9.2 day 2 Maclaurin Series Liberty Bell, Philadelphia, PA

9.2 Taylor Series.

Chapter 10 sequences Series Tests find radius of convg R

Sum and Difference Formulas (Section 5-4)

TAYLOR SERIES Maclaurin series ( center is 0 )

Finding Common Maclaurin Series

Presentation transcript:

how to represent certain types of functions as sums of power series MACLAURIN SERIES how to represent certain types of functions as sums of power series You might wonder why we would ever want to express a known function as a sum of infinitely many terms. Integration. (Easy to integrate polynomials) Finding limit Finding a sum of a series (not only geometric, telescoping) How does the calculator find values of sine (or cosine or tangent)?

Maclaurin series ( center is 0 ) Example: Maclaurin series ( center is 0 ) Example: Find Maclaurin series

MACLAURIN SERIES Example: Find Maclaurin series Example: Find Maclaurin series Example: Find Maclaurin series Example: Find Maclaurin series Example: Find Maclaurin series

MACLAURIN SERIES Important Maclaurin Series and Their Radii of Convergence MEMORIZE: ** Students are required to know the series listed in Table 10.1, P. 620 How to memorize them

Important Maclaurin Series and Their Radii of Convergence MEMORIZE: ** Students are required to know the series listed in Table 10.1, P. 620 Denominator is n! even, odd Denominator is n odd

Maclaurin series ( center is 0 ) How to find a Maclaurin Series of a function Use the formula Use the known functions 1) Replace each x 2) Diff 3) integrate 3) Find a product between two

MACLAURIN SERIES TERM-081

MACLAURIN SERIES TERM-091

MACLAURIN SERIES TERM-101

MACLAURIN SERIES TERM-082

MACLAURIN SERIES TERM-102

MACLAURIN SERIES TERM-091

TAYLOR AND MACLAURIN Example: Find the sum of the series

MACLAURIN SERIES TERM-102

MACLAURIN SERIES TERM-082

MACLAURIN SERIES TERM-131

MACLAURIN SERIES Example: Find the sum Leibniz’s formula:

Important Maclaurin Series and Their Radii of Convergence MEMORIZE: ** Students are required to know the series listed in Table 10.1, P. 620 Denominator is n! even, odd Denominator is n odd MAC

Example: MACLAURIN SERIES Important Maclaurin Series and Their Radii of Convergence Example: Find Maclaurin series

MACLAURIN SERIES TERM-122

MACLAURIN SERIES TERM-082

Important Maclaurin Series and Their Radii of Convergence MEMORIZE: ** Students are required to know the series listed in Table 10.1, P. 620 Denominator is n! even, odd Denominator is n odd