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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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