By: Anna Levina edited: Rhett Chien. Section 10.1 Maclaurin and Taylor polynomial Approximations...
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Transcript of By: Anna Levina edited: Rhett Chien. Section 10.1 Maclaurin and Taylor polynomial Approximations...
![Page 1: By: Anna Levina edited: Rhett Chien. Section 10.1 Maclaurin and Taylor polynomial Approximations Recall: Local Linear Approximation Local Quadratic (Cubic)](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c755503460f94928d02/html5/thumbnails/1.jpg)
by: Anna Levina
edited: Rhett Chien
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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
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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
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• 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.
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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
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Visualization
• y = e^x• linear: y = 1 + x• quadratic:
y x 2
2 x 1
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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.
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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
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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
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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
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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
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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.
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Just a cool picture
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Section 10.2Sequences
• Definition of a sequence
• Limit of a sequence
• The squeezing theorem for sequences
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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,…
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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
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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→+∞.
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Example
The general term for the
sequence 3, 3/8, 1/9, 3/64,… is
3/n^3
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Example
Show that +∞{ln(n)/n} converges. n=1 What is the limit?
0
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Section 10.3Monotonic sequences
• Strictly monotonic
• Monotonic
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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 ≥ …
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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
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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.
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Example
Determine which answer best describes the sequenced
+∞{6/n} n=1
A. Strictly increasing B. Strictly decreasing
C. Increasing D. Decreasing
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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
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Sum
= a0 + a1 + a2 +…. an
Partial sum
Sn = a1 + a2 + ... + an, the nth partial sum.
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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.
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Geometric Series
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Telescoping sums
A sum in which subsequent terms cancel each other, leaving only initial and final terms. For example,
S= =
=
=
is a telescoping sum.
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Harmonic Series
• Always diverge
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Example
• The sum
is convergent with sum 1.
Sn = = - = 1 - 1 as n
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Section 10.5Convergence tests
• Divergence test
• Integral test
• P-series
• The comparison test
• The limit comparison test
• The ratio test
• Root test
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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.
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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
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p-Series
• The series
is called a p-Series.
if p > 1 the p-series converges
if p ≤ 1 the p-series diverges
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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
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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.
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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
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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
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Example
• The series
is called Euler's series. It converges to Euler's number e.
Does Euler's series converge ?
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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.
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Example
• Determine if the following series is
convergent or divergent
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.
Since we conclude, from the Ratio-Test, that the series
is convergent.
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Example
Determine whether series converges and find the sum.
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Section NextAlternating Series. Conditional Convergence
• Alternating Series
• AST
• Absolute Convergence
• Conditional Convergence
• The Ratio Test for absolute convergence
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Alternating Series
• A series of a form or
Where
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Alternating Series Test
• Also known as the Leibniz criterion. An alternating series converges if
and
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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.
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Conditional Convergence
• If the series converges, but
does not, where is the absolute value, then the series is said to be conditionally convergent.
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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.
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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.
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Section 10.8Maclaurin and Taylor Series;
Power series
• Maclaurin and Taylor series
• Power series
• Radius and interval of convergence
• Function defined by power series
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Taylor Series
If f has derivatives of all orders xo, then we call the series
the Taylor Series for f about x=x0.
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Maclaurin Series
A Maclaurin series is a Taylor series
of a function f about 0
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Example
Find the Taylor series with
center x=x0 for
so f(0)=1.
so f'(0)=0
so
so f'''’(0)=0.
so
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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.
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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.
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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
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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.
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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
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Example
Find the radius of convergenceThe general term of the series has the form
Consequently, the radius of convergence equals 1
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• Measuring infinity
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Section 10.9Convergence of Taylor Series
• The nth remainder
• Estimating the nth remainder
• Approximating different functions
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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.
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Lagrange Remainder
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Section 10.10differentiating and integrating power series
• Differentiating power series
• Integrating power series
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Differentiation
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Integration
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Examples online
http://archives.math.utk.edu/visual.calculus/6/power.2/index.html
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Bibliography• http://mathworld.wolfram.com/• http://www.fractalzone.be/picture.php?img=120&res=med• http://www.armchair.com/aware/aging1a.html• http://www.troutmusic.com/kids.htm• http://adsoftheworld.com/media/print/bic_infinity• http://www.math.yorku.ca/infinity/ExamSales/ExamSales.html• http://fusionanomaly.net/infinity.html• http://www.lincolnseligman.co.uk/infinity.htm• http://www.primarystuff.co.uk/photos/displayimage.php?album=topn&cat=0&pos=0• http://sprott.physics.wisc.edu/fractals/collect/1995/• http://www.tcdesign.net/infinite_possibility.htm• http://www.irtc.org/stills/1999-06-30/view.html• http://fusionanomaly.net/fractals.html• http://apophysisrocks.wordpress.com/• http://www.piglette.com/fractals/fractaluniverse.html• http://www.lactamme.polytechnique.fr/Mosaic/images/VONK.32.D/display.html• http://archives.math.utk.edu/visual.calculus/• http://tutorial.math.lamar.edu/AllBrowsers/2414/RatioTest.asp• http://www.sosmath.com/calculus/series/rootratio/rootratio.html• Anton, Bivens, Davis. Calculus. Anton textbooks, 2002.
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