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Power Series

Power Series. Dr. Dillon Calculus II Fall 1999. Recall the Taylor polynomial of degree. for a function. times differentiable at. which is. Taylor Polynomials. The Taylor series for a function. which has derivatives of all orders at a point. Definition. is given by. Compare.

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Power Series

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  1. Power Series Dr. Dillon Calculus II Fall 1999

  2. Recall the Taylor polynomial of degree for a function times differentiable at which is Taylor Polynomials

  3. The Taylor series for a function which has derivatives of all orders at a point Definition is given by

  4. Compare Taylor polynomial Taylor series

  5. A Taylor Polynomial is a (finite) sum with a defined degree.

  6. A Taylor Series is an infinite sum, i.e., a sequence of partial sums.

  7. Sequence of Partial Sums Each member of the sequence of partial sums is itself a Taylor polynomial. A Taylor series is a sequence of Taylor polynomials.

  8. Let Sequence of Taylor Polynomials

  9. at to be the sequence As the Taylor Series We actually define the Taylor series for

  10. at An Example The first few Taylor polynomials for The Taylor series:

  11. you need derivatives up to order Assumptions For a Taylor polynomial, For a Taylor series, you need derivatives of all orders.

  12. at at at at Examples We could find Taylor series for

  13. at for the Taylor series for Notation Using sigma notation write

  14. at The Taylor series for is called the Maclaurin series for Terminology

  15. at Will the Calculator Find a Taylor Series? Maybe; try it for

  16. Where Does One Start? Start with a Taylor polynomial. Try using a large degree. Can you see the pattern? I can’t!

  17. Find the coefficient of the degree term at derivative of Start by finding the Now What? in the Taylor series for

  18. Use the Calculator to... get successive expressions for

  19. The Pattern? Look at a couple more: ...

  20. is good for and then Conclusion

  21. When

  22. The coefficient for the degree term is at Thus in the Taylor series for

  23. Notice Finding a Taylor series means finding the coefficients.

  24. Taylor Series Degree n term Coefficient of Degree n Term

  25. Separate Issues • What is the Taylor series for a function at a point? • For what values of x does the Taylor series for a function converge? • Does the Taylor series for a function converge to that function?

  26. Let are all at A Weird Example It’s hard to see that it even exists, but Thus the coefficients in the Taylor series for

  27. Meaning... This Taylor series describes the function well but only at one point, 0. In cases like this, Taylor series aren’t good for much.

  28. Lucky for Us All of the usual suspects can be well represented by their Taylor series at all points where they are infinitely differentiable. The Taylor series for all of our favorite functions converge to the functions at least on a decent sized interval, if not on the entire real line.

  29. Meaning... • For all algebraic functions, • for trigonometric functions and their inverses, • for exponential functions and logarithms, there are excellent polynomial approximations to the functions on intervals surrounding points where the functions have derivatives of all orders.

  30. that is to within two decimal places of on the interval (2,6). Use a Taylor polynomial at Example Find a polynomial approximation to Solution?

  31. What Degree Do We Need? Use your technology to figure it out!

  32. How Can You Tell? If the polynomial and the function agree at the endpoints, they agree at all the points between. That’s a Big Theorem.

  33. A power series at is a function of the form The Power Series are the coefficients.

  34. is a power series at Example All the coefficients are 1.

  35. Recall

  36. If for some positive number then Huge Theorem is the Taylor series for f at a.

  37. So... is the Maclaurin series for

  38. More Interesting Stuff If a power series converges on an interval, we can • differentiate term by term to get another convergent power series • integrate term by term to get another convergent power series • take limits term by term, on the interval of convergence • do arithmetic term by term to get still more convergent power series.

  39. converges to Example In other words, for a given value of x

  40. for means we can substitute True For All x... That gives us which is the Maclaurin series for that function.

  41. More Cute Tricks means

  42. Find the Maclaurin series for Start with the Maclaurin series for It converges to One Last Example Solution

  43. This is the Maclaurin series for Thus... It converges to the function on the whole real line.

  44. A Final Note Memorize the following series: • The Maclaurin series for sine and cosine • The Maclaurin series for the natural exponential The Maclaurin series for 1/(1-x)

  45. Be Able to Use Them to find Taylor series for functions obtained from the above via • Differentiation • Integration Arithmetic

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