12'8 Power Series

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12.8 Power Series

• Math 6B
• Calculus II

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Power Series

• Power Series centered at x 0.
• Power Series centered at x a.

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How to Test a Power Series for Convergence

• Use the Ratio test (or Root test) to find the
  interval where the series converges absolutely.
  Ordinarily this is an open interval or a
  R lt x lt a R
• If the interval of absolute convergence is
  finite, test for convergence or divergence at
  each endpoint. Use an appropriate test.

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How to Test a Power Series for Convergence

• If the interval of absolute convergence is a
  R lt x lt a R, the series diverges for

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The Radius and Interval of Convergence

• Possible behavior of
• There is a positive number R (also known as the
  radius of convergence) such that the series
  diverges for all but converges for
  . The series may or may not
  converge at either endpoint x a R
  and x a R.

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The Radius and Interval of Convergence

• The series converges absolutely for every x
• The series converges at x a and diverges
  everywhere else. (R 0 )