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(MTH 250)

(MTH 250). Calculus. Lecture 16. Previous Lecture’s Summary. Subsequences Limit superior and limit inerior Cauchy sequences Infinite s eries Convergence of infinite series. Absolutely convergent series. Today’s Lecture. Recalls Polynomial approximations Maclaurin polynomials

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(MTH 250)

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  1. (MTH 250) Calculus Lecture 16

  2. Previous Lecture’s Summary • Subsequences • Limitsuperior and limitinerior • Cauchy sequences • Infiniteseries • Convergence of infinite series. • Absolutely convergent series

  3. Today’s Lecture • Recalls • Polynomial approximations • Maclaurin polynomials • Taylor polynomials • Taylor and Maclaurin series • Convergence • Some important series

  4. Recalls Theorem: Suppose Then every subsequence of also converges to , that is Corollary: Suppose has two subsequences that converge to different limits. Then is divergent. Let Limit superior: Let be a sequence. The limit superior of denoted by , is defined to be the i.e.

  5. Recalls Let Limit inferior: Let be a sequence. The limit inferior of denoted by , is defined to be the i.e. Definition: A sequence is called a Cauchy sequence if given any there exists a natural number such that for all we have . Theorem: (Bolzano-Weierstrass theorem) Every bounded sequence has a convergent subsequence

  6. Recalls Proposition: Every Cauchy sequence is bounded. Theorem: (Cauchy criterion) A sequence is convergent iff it is a Cauchy sequence. Definition: An infiniteseriesis an expression thatcanbewritten in the form The numbers are called the terms of the series

  7. Recalls Definition:Let denote the sum of the initial terms of the series up to and including the termwith index The numbers are called the partial sum of the series and the sequence is called the sequence of partial sums.

  8. Recalls Definition: E

  9. Recalls Geometric series: A series of the form … is called the geometric series. Theorem

  10. Recalls Example: The series converges if . The expanded form of the series is Since, it is a geometric series with it converges to i.e.

  11. Recalls Definition:

  12. Polynomial approximations Local linear approximation: The approximation polynomial: is first degree polynomial with property: and If the curve has a pronounced bend the local linear approximation would not be appropriate.

  13. Polynomial approximations For we try to find a local quadratic approximation of of the form . The and are to be chosen in such a way that the approximation polynomial: satisfies: But and Therefore

  14. Polynomial approximations Problem:Given a function times differentiable, find a polynomial of degree of the form with property: Answer: By definition Therefore, .

  15. Polynomial approximations

  16. Maclaurin Polynomials Definition:If can be times differentiated at 0, then we define the Maclaurinpolynomial for to be Remark: By definition the Maclaurin polynomial has the property that its value and the values of its first derivatives match the values of and its first derivatives at . Any smooth function can be approximated as a polynomial. This representation provides a means to predict the value of a function at one point in terms of the function value and its derivatives at another point.

  17. Maclaurin Polynomials Example: Solution:

  18. Maclaurin Polynomials Cont..

  19. Maclaurin Polynomials Example: Solution: Thus:

  20. Maclaurin Polynomials Cont. The graph of and four of its polynomial approximations .

  21. Maclaurin Polynomials Example:Find the Maclaurin polynomial for Solution: If

  22. Maclaurin Polynomials Cont: Consequently,

  23. Taylor Polynomials • Uptill now, we have approximatedfunctionsaround. • Suppose there is a point , we can also approximate a function in the vicinity of . • Instead of expressing in the powers of , we express it using powers of • where • We call it the Taylor polynomial of degree

  24. Taylor Polynomials Definition:If can be times differentiated at , then we define the Taylorpolynomial for to be . • Moreover, for • Maclaurin polynomicalis the Taylor polynomial with

  25. Taylor Polynomials Example: Solution:

  26. Taylor Polynomials

  27. Taylor Polynomials Remainder: As wedefine the remainder as the differencebetween and its Taylor polynomial, that is It gives us the indication of the accuracy of the approximation. Theorem (Lagrange formula for the remainder): The remainder in Taylor polynomial approximation of degree n can be written as with some number between and .

  28. Taylor & Maclaurin Series Definition:

  29. Taylor & Maclaurin Series Example: Maclaurin seriesrepresentation of Solution:

  30. Taylor & Maclaurin Series Cont:

  31. Taylor & Maclaurin Series Example: Find the value of given that and all other higher order derivatives of at are zero. Solution: By Taylor series and and . Since the higherorderderivatives are zero

  32. Taylor & Maclaurin Series Examples: Find the Maclaurin polynomial for f(x) = sinx Solution Therefore:

  33. Taylor & Maclaurin Series Example: Solution:

  34. Taylor & Maclaurin Series Example: Find the Maclaurin Series of Solution:

  35. Taylor & Maclaurin Series Example: Find the Maclaurin Series of Solution: This is a geometric series with and

  36. Taylor & Maclaurin Series Example: Find the Maclaurin polynomial for f(x) = cos2 x. Solution: Since Write the Maclaurinseries for cos x and replace x by 2x, and then simplify. The Maclaurin series of cos x is

  37. Convergence • Taylor serieswork very well for polynomials; the exponential function ex and the sine and cosine functions. (They are all examples of entire functions –i.e., f(x) equals its Taylor series everywhere). • Taylor series do not always work well. For example, for the logarithm function, the Taylor series do not converge if x is far from x0. For example: Log approximation around 0:

  38. Convergence Theorem: For any Taylor seriesin exactly one of the followingistrue: • The series converges only for . • The series converges absolutely (and hence converges for all real values of . • The series converges absolutely (and hence converges for all x in somefinite open intervaland diverges if or . Ateither of the values or , the seriesmay converge or diverge, depending on the particularseries.

  39. Let Convergence Example: It’s hard to see that it even exists, but at and are all 0. Thus the coefficients in the Taylor series for

  40. Convergence Example: What is the interval of convergence? Solution: Since , the series converges , or . In interval notation Test endpoints of –1 and 1. Series diverges Series diverges

  41. Some important series

  42. Lecture Summary • Recalls • Polynomial approximations • Maclaurin polynomials • Taylor polynomials • Taylor and Maclaurin series • Convergence • Some important series

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