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by: Anna Levina edited: Rhett Chien

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Section 10.1 Maclaurin 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 x 0 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 x 0. If the graph of the function f(x) has a pronounced “bend” at x 0, then we can expect that the accuracy of the local linear approximation of f at x 0 will decrease rapidly as we progress away from x 0. The way to deal with this problem is to approximate the function f at x 0 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 x 0. As a result, we can expect that the graph of p will remain closer to the graph of f over a larger interval around x 0 than the graph of the local linear approximation. Polynomial p is local quadratic approximation of f at x=x 0.

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Local Quadratic Approximation f(x) ≈ ax^2 + bx + c let x 0 = 0 f(x 0 ) = f(0) = 0 f’(x) = 2ax + b f’(x 0 ) = f’(0) = b f”(x) = 2a f”(x 0 ) = f”(0) = 2a to find a, b, c: f(0) = c f’(0) = b f”(0)/2 = a

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

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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

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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 p 2 (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 p 2 = 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 10.2 Sequences 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 {a n } n=1 to be an alternative notation for the function f(n) = a n, 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 {a n }, {b n }, and {c n }, be sequences such that a n ≤ b n ≤c n If sequences {a n } and {c n } have a common limit L as n→ +∞, then {b n } 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 10.3 Monotonic sequences Strictly monotonic Monotonic

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Definition A sequence {a n } n=1 is called Strictly increasing if a 1 < 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 a n+1 > a n Method 3 (Ratio) a n+ 1 / a n < 1 Method 4 a n = 1/n Let f(x)=1/x f’(x)= -x^(-2) f’(x) < 0 for all x≥ 1 Therefore a n is strictly decreasing

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Eventually 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.

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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 10.4 Infinite Series 1.Sums of infinite series 2.Geometric series 3.Telescoping sums 4.Harmonic series

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Sum = a0 + a1 + a2 +…. an Partial sum Sn = a1 + a 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. S n = = -= 1 -1 as n

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Section 10.5 Convergence 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 | b n | | a n | 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 seriesconverges 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 Next Alternating 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 10.8 Maclaurin 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 x o, then we call the series the Taylor Series for f about x=x 0.

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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=x 0 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 c n 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 representedIf a function f is expressed as a power series on some interval, then we say that f is represented by the power series on the interval.

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Some series to remember = = = = = Dollars equal cents Theorem: 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 convergence The general term of the series has the form C onsequently, the radius of convergence equals 1

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Measuring infinity

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Section 10.9 Convergence 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 differentiating 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

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Bibliography Anton, Bivens, Davis. Calculus. Anton textbooks, 2002.

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