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Representation of Functions by Power Series

Representation of Functions by Power Series. Lesson 9.9. Geometric Power Series. Consider the function Note the similarity to the sum of the geometric series . Geometric Power Series. Thus we could let a = 1 and r = x and have Where is this power series centered?

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Representation of Functions by Power Series

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  1. Representation of Functions by Power Series Lesson 9.9

  2. Geometric Power Series • Consider the function • Note the similarity to the sum of the geometric series

  3. Geometric Power Series • Thus we could let a = 1 and r = x and have • Where is this power series centered? • At x = 0, for the interval -1 < x < 1 • Note also that our original f(x) is defined for all x ≠ 1

  4. Geometric Power Series • To center this series at x = -3 you could specify • Then • a = ¼ • r = [(x + 3)/4] • So convergence would be for the interval

  5. Geometric Power Series • Try this out … find a power series for the function centered at c

  6. Operations with Power Series • Note the following rules

  7. Adding Two Power Series • Consider a function which can be broken into two using partial fractions • Now determine two geometric power series which are added to give us a series for the original function

  8. Try Another • Try these … just for practice See Example 4 and Example 2

  9. Assignment • Lesson 9.9 • Page 674 • Exercises 5 – 25 odd

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