by: Anna Levina

edited: Rhett Chien

Section 10.1Maclaurin and Taylor polynomial

Approximations

• Recall: Local Linear Approximation

• Local Quadratic (Cubic) Approximation

• Maclaurin Polynomials

• Taylor Polynomials

• Sigma Notation for Taylor and Maclaurin Polynomials

• The nth Remainder

Local Linear Approximation

Local linear approximation of a function f at x0 is

• f(x) = e^x

• Tangent:

• y = 1+x

Local linear

approximation

f(x) ≈ 1+x

• Linear approximation works only on values close to x0.• If the graph of the function f(x) has a pronounced

“bend” at x0, then we can expect that the accuracy of the local linear approximation of f at x0 will decrease rapidly as we progress away from x0.

• The way to deal with this problem is to approximate the function f at x0 by a polynomial p of degree 2 with the property that the value of o and the values of its first two derivatives match those of f at x0. As a result, we can expect that the graph of p will remain closer to the graph of f over a larger interval around x0 than the graph of the local linear approximation.

• Polynomial p is local quadratic approximation of f at x=x0.

Local Quadratic Approximation

f(x) ≈ ax^2 + bx + clet x0 = 0f(x0) = f(0) = 0f’(x) = 2ax + bf’(x0) = f’(0) = bf”(x) = 2af”(x0) = f”(0) = 2ato find a, b, c:f(0) = cf’(0) = bf”(0)/2 = a

Visualization

• y = e^x• linear: y = 1 + x• quadratic:

y x 2

2 x 1

Maclaurin Polynomials

The accuracy of the approximation increases as the degree of the polynomial increases.

We use Maclaurin polynomial.

If f can be differentiated n times at 0, then we define the nth Maclaurin polynomial to be

( )2 3''(0) '''(0) (0)

( ) (0) '(0) ...2! 3! !

nnf f f

f x f f x x x xn

around x 0.

Taylor Polynomials

If f can be differentiated n times at a, then we define the

nth Tylor polynomial for f about x = a to be

Example• Find the Maclaurin polynomial

of order 2 for e^(3x)

f(0) = 1 = c

f’(0) = 3 = b

f”(0) = 9 = 2a

p2(x) = 1 + 3x + 9(x^2)/2

Example

Find a Taylor polynomial for f(x) =

3lnx of order 2 about x=2

f(2) = 3ln2

f’(2) = 3/2

f” (2) = - 3/4

p2 = 3ln2 + 3/2(x-2) – 3/8(x-2)^2

Sigma Notation for Taylor and Maclaurin Polynomials

We can write the nth-order Maclaurin polynomial for f(x) as

We can write the nth-order Taylor polynomial for f(x) about c as

The nth Remainder• If the function f can be differentiated n+1 times on an interval

I containing the number x0, and if M is an upper bound for

on I, ≤ M for all x in I, then

for all x in I.

Just a cool picture

Section 10.2Sequences

• Definition of a sequence

• Limit of a sequence

• The squeezing theorem for sequences

Definition of a Sequence

A sequence is a function whose domain is a

set of integers. Specifically, we will regard

the expression {an}n=1 to be an alternative

notation for the function

f(n) = an, n=1,2,3,…

Limit of a sequence

A sequence {an} is said to converge to the limit L if given any ε>0, there is a positive integer N such that Ian – LI < ε for n≥N. In this case we write

A sequence that does not converge to some finite

limit is said to diverge

The squeezing theorem for sequences

• Let {an}, {bn}, and {cn}, be sequences such that an≤ bn≤cn

• If sequences {an} and {cn} have a common limit L as n→ +∞, then {bn} also has the limit L as n→+∞.

Example

The general term for the

sequence 3, 3/8, 1/9, 3/64,… is

3/n^3

Example

Show that +∞{ln(n)/n} converges. n=1 What is the limit?

0

Section 10.3Monotonic sequences

• Strictly monotonic

• Monotonic

Definition

• A sequence {an}n=1 is called• Strictly increasing if a1 < a2 < a3 < … < an< …

• Increasing if a1≤ a2≤ a3 ≤ … an ≤ …

• Strictly decreasing if a1 > a2 > a3 > … an > …

• Decreasing if a1≥ a2 ≥ a3 ≥ … an ≥ …

Testing for Monotonicity

Method 1.

By inspection

Method 2

an+1 > an

Method 3 (Ratio)

an+1/an< 1

Method 4

an = 1/nLet f(x)=1/xf’(x)= -x^(-2)f’(x) < 0 for all x≥ 1

Therefore an is strictly decreasing

Eventually

• If discarding infinitely many terms from a beginning of a sequence produces a sequence with certain property, then the original sequence is said to have that property eventually.eventually.

Example

Determine which answer best describes the sequenced

+∞{6/n} n=1

A. Strictly increasing B. Strictly decreasing

C. Increasing D. Decreasing

Section 10.4Section 10.4Infinite SeriesInfinite Series

1.1. Sums of infinite seriesSums of infinite series

2.2. Geometric seriesGeometric series

3.3. Telescoping sumsTelescoping sums

4.4. Harmonic seriesHarmonic series

Sum

= a0 + a1 + a2 +…. an

Partial sum

Sn = a1 + a2 + ... + an,    the nth partial sum.

Convergent series

• If Sn exists, we say that an is a convergent series, and write

Sn = an.• Thus a series is convergent if and only if

it's sequence of partial sums is convergent. The limit of the sequence of partial sums is the sum of the series. A series which is not convergent, is a divergent series.

Geometric Series

Telescoping sums

A sum in which subsequent terms cancel each other, leaving only initial and final terms. For example,

S= =

=

=

is a telescoping sum.

Harmonic Series

• Always diverge

Example

• The sum

is convergent with sum 1.

Sn = = - = 1 - 1    as    n

Section 10.5Convergence tests

• Divergence test

• Integral test

• P-series

• The comparison test

• The limit comparison test

• The ratio test

• Root test

Divergence test

If the series converges, then the

sequence converges to zero. Equivalently:

If the sequence does not converge to

zero, then the series

can not converge.

Integral test

• Suppose that f(x) is positive, continuous, decreasing function on the interval [N, ). Let a n = f(n). Then

       converges if and only if           converges

p-Series

• The series

is called a p-Series.

if p > 1 the p-series converges

if p ≤ 1 the p-series diverges

Comparison test

• Suppose that converges absolutely, and is a sequence of numbers for which

| bn | | an | for all n > N

If the series converges to positive infinity, and

is a sequence of numbers for which for all n > N

also diverges.

Then the series converges absolutely as well.

Then the series

Limit Comparison Test

• Suppose and are two infinite

series. Suppose also that r = lim | a n / b n | exists, and 0 < r <

Then converges absolutely if and only

if converges absolutely.

Ratio test

• Consider the series . Then

•if lim | a n+1 / a n | < 1 then the series converges absolutely.

n•if there exists an N such that | a n+1 / a n |   1 for all n > N then the series diverges.

•if lim | a n+1 / a n | = 1, this test gives no information

n

Root test

• Consider the series . Then:

• if lim sup | a n |^ (1/n) < 1 then the series converges absolutely.

• if lim sup | a n |^ (1/n) > 1 then the series diverges • if lim sup | a n |^ (1/n) = 1, this test gives no

information

Example

• The series

is called Euler's series. It converges to Euler's number e.

Does Euler's series        converge ?

Example

• Does the series

converge or diverge ?

We will use the limit comparison test, together with the p-series test. First, note that 1 / (1 + n^ 2) < 1 / n^ 2

But since the series •   1 / n 2 is a p-series with p = 2, and therefore converges,

the original series

must also converge by the comparision test.

Example

• Determine if the following series is

convergent or divergent

                                              .

Since we conclude, from the Ratio-Test, that the series

is convergent.

Example

Determine whether series converges and find the sum.

Section NextAlternating Series. Conditional Convergence

• Alternating Series

• AST

• Absolute Convergence

• Conditional Convergence

• The Ratio Test for absolute convergence

Alternating Series

• A series of a form or

Where

Alternating Series Test

• Also known as the Leibniz criterion. An alternating series converges if

and                 

Absolute convergence

• A series is said to converge absolutely if the series converges, where denotes the absolute value.

• If a series is absolutely convergent, then the sum is independent of the order in which terms are summed.

• Furthermore, if the series is multiplied by another absolutely convergent series, the product series will also converge absolutely.

Conditional Convergence

• If the series converges, but

does not, where is the absolute value, then the series is said to be conditionally convergent.

The ration test for absolute convergence

• The same as ratio test, just use absolute value.

• If the series diverges absolutely, check for conditional convergence using another method.

Example

Classify the series as either absolutely

convergent, conditionally convergent, or

divergent.

by the Alternating Series Test, the series

is convergent. Note that it is not absolutely convergent.

Section 10.8Maclaurin and Taylor Series;

Power series

• Maclaurin and Taylor series

• Power series

• Radius and interval of convergence

• Function defined by power series

Taylor Series

If f has derivatives of all orders xo, then we call the series

the Taylor Series for f about x=x0.

Maclaurin Series

A Maclaurin series is a Taylor series

of a function f about 0

                                                                                                    

Example

Find the Taylor series with

center x=x0 for

so f(0)=1.

so f'(0)=0

so

so f'''’(0)=0.

so

Power Series

A power series about a, or just power series, is any series that

can be written in the form, where a and cn are numbers.

Radius and interval of convergence

For any power series in x, exactly one of the following is true:a. The series converges only for x=0.b. The series converges absolutely (and hence

converges) for all real values of x.c. The series converges absolutely (and hence

converges) for all x in some finite open interval (-R,R). At either of the values x=R or x=-R, the series may converge absolutely, converge conditionally, or diverge, depending on the particular series.

Finding the interval of convergence

Use ratio test for absolute convergence

If p = then the series

is convergent.

Find values of IxI for which p<1

Function defined by power series

• If a function f is expressed as a power series on some interval, then we say that f is representedrepresented by the power series on the interval.

Some series to remember

=

=

=

=

=

Dollars equal centsTheorem: 1$ = 1c.Proof:And another that gives you a sense of money disappearing.

1$ = 100c= (10c)^2= (0.1$)^2= 0.01$= 1c

Example

Find the radius of convergenceThe general term of the series has the form

Consequently, the radius of convergence equals 1

• Measuring infinity

Section 10.9Convergence of Taylor Series

• The nth remainder

• Estimating the nth remainder

• Approximating different functions

The nth Remainder

Problem.Given a function f that has derivatives of all

orders at x = x0, determine whether there is an open interval containing x0 such that f(x) is the sum of its Taylor series about x=x0 at each number in the interval; that is

for all values of x in the interval.

Lagrange Remainder                                                                                                                       

Section 10.10differentiating and integrating power series

• Differentiating power series

• Integrating power series

Differentiation

Integration

Examples online

http://archives.math.utk.edu/visual.calculus/6/power.2/index.html

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