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Section 8.7: Power Series

Section 8.7: Power Series. Definition is a power series centered at c. If c = 0,. IMPORTANT: A power series is a function . Its value and whether or not it converges depends on which x you plug in. Think of them as “ Long Polynomials ”.

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Section 8.7: Power Series

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  1. Section 8.7: Power Series

  2. Definition is a power series centered at c If c = 0, IMPORTANT: A power series is a function. Its value and whether or not it converges depends on which x you plug in. Think of them as “Long Polynomials”

  3. If you plug in a number for x, you get a geometric series. It converges if |x|<1. Remember, a power series is a function.

  4. Find all values of x at which the power series converges. By ratio test the series converges absolutely when

  5. If x-5 is positive: If x-5 is negative:

  6. Let’s try geometry instead of algebra. |x-5| is really just the distance from x to 5 2 2 3 5 7 The Radius of Convergence is 2.

  7. By ratio test converges absolutely when 3 < x < 7. But we must still check endpoints because the ratio test tells us nothing at x = 3 and x = 7. Diverges Converges Radius of convergence R = 2 Interval of convergence [3, 7)

  8. Try the Root Test this time. Unless x = 2 So the series converges only at x = 2 Radius of convergence is 0

  9. So the series converges no matter what x is. Radius of convergence = Interval of convergence=

  10. Theorem For a power series exactly one of these possibilities occurs: • It converges only at x = c • It converges at every number x • There is a number R > 0 so that • It converges absolutely if |x-c|<R • It diverges if |x-c|>R

  11. Suppose converges at x = 8 Does it converge at x = 6 ? Yes ! Does it converge at x = -1 ? Yes ! Maybe ? Does it converge at x = -2 ? Maybe ? Does it converge at x = 9? -2 3 8

  12. The Calculus of Power Series If a power series is a function, can we integrate or differentiate it? How much like a polynomial is it?

  13. Theorem This is valid on the interval of convergence of the original series.

  14. So what is ?? It must be ex!!

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