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Mastering Taylor and Maclaurin Series: A Comprehensive Guide

Learn how to represent functions as power series, integrate polynomials, find limits, and sums of series using Taylor and Maclaurin series. Understand important Maclaurin Series, Binomial series, Taylor polynomials, and their applications with examples.

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Mastering Taylor and Maclaurin Series: A Comprehensive Guide

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  1. TAYLOR AND MACLAURIN • how to represent certain types of functions as sums of power series • You might wonder why we would ever want to express a known function as a sum of infinitely many terms. • Integration. (Easy to integrate polynomials) • Finding limit • Finding a sum of a series (not only geometric, telescoping)

  2. TAYLOR AND MACLAURIN Example: Maclaurin series ( center is 0 ) Example: Find Maclaurin series

  3. TAYLOR AND MACLAURIN Important Maclaurin Series and Their Radii of Convergence MEMORIZE: these Maclaurin Series

  4. TAYLOR AND MACLAURIN Maclaurin series ( center is 0 ) Example: Find Maclaurin series

  5. TAYLOR AND MACLAURIN TERM-081

  6. TAYLOR AND MACLAURIN TERM-091

  7. TAYLOR AND MACLAURIN TERM-101

  8. : TAYLOR AND MACLAURIN TERM-082

  9. Sec 11.9 & 11.10: TAYLOR AND MACLAURIN TERM-102

  10. Sec 11.9 & 11.10: TAYLOR AND MACLAURIN TERM-091

  11. TAYLOR AND MACLAURIN Maclaurin series ( center is 0 ) Example: Find the sum of the series

  12. TAYLOR AND MACLAURIN TERM-102

  13. TAYLOR AND MACLAURIN TERM-082

  14. TAYLOR AND MACLAURIN Example: Find the sum Leibniz’s formula:

  15. TAYLOR AND MACLAURIN Important Maclaurin Series and Their Radii of Convergence MEMORIZE: these Maclaurin Series

  16. The Binomial Series DEF: Example: Example:

  17. The Binomial Series binomial series. NOTE:

  18. The Binomial Series TERM-101 binomial series.

  19. The Binomial Series TERM-092 binomial series.

  20. TAYLOR AND MACLAURIN Important Maclaurin Series and Their Radii of Convergence Example: Find Maclaurin series

  21. TAYLOR AND MACLAURIN TERM-102

  22. TAYLOR AND MACLAURIN TERM-111

  23. TAYLOR AND MACLAURIN TERM-101

  24. TAYLOR AND MACLAURIN TERM-082

  25. TAYLOR AND MACLAURIN Important Maclaurin Series and Their Radii of Convergence MEMORIZE: these Maclaurin Series

  26. TAYLOR AND MACLAURIN Maclaurin series ( center is 0 ) Taylor series ( center is a )

  27. TAYLOR AND MACLAURIN TERM-091

  28. TAYLOR AND MACLAURIN TERM-092

  29. TAYLOR AND MACLAURIN TERM-082

  30. TAYLOR AND MACLAURIN Taylor series ( center is a ) DEF: Taylor polynomial of order n

  31. TAYLOR AND MACLAURIN The Taylor polynomial of order 3 generated by the function f(x)=ln(3+x) at a=1 is: TERM-102 DEF: Taylor polynomial of order n

  32. TAYLOR AND MACLAURIN TERM-101

  33. TAYLOR AND MACLAURIN TERM-081

  34. TAYLOR AND MACLAURIN Taylor series ( center is a ) Taylor polynomial of order n Remainder Taylor Series Taylor’s Formula Remainder consist of infinite terms for some c between a and x. REMARK: Observe that :

  35. TAYLOR AND MACLAURIN Taylor’s Formula for some c between a and x. Taylor’s Formula for some c between 0 and x.

  36. TAYLOR AND MACLAURIN Taylor series ( center is a ) DEF: nth-degree Taylor polynomial of f at a. DEF: Remainder Example:

  37. TAYLOR AND MACLAURIN TERM-092

  38. TAYLOR AND MACLAURIN TERM-081

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