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9.8 Power Series

9.8 Power Series. Power Series. A power series centered at x = a is given by A power series is a generalized version of a geometric series. Use Ratio Test to check the convergence of a power series. Example : Determine the convergence of the following series. .

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9.8 Power Series

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  1. 9.8Power Series

  2. Power Series A power seriescentered at x = a is given by A power series is a generalized version of a geometric series. Use Ratio Test to check the convergence of a power series. Example: Determine the convergence of the following series.

  3. Interval & Radius of Convergence Given the power series There are 3 possibilities: • The series converges only when x = a Then the radius of convergence R = 0 • The series converges for all x Then the radius of convergence R = ∞ 3) There is a positive number R such that the series converges if |x - a| < R and diverges if |x - a| > R We still must check the endpoints The interval on which a series converges is called the interval of convergence.

  4. for all . At , the series is , which converges to zero. Example Find the interval of convergence and the radius of convergence for the series Radius R = 0

  5. Check at endpoints to conclude that The interval of convergence is (0,2]. Example Find the interval of convergence and the radius of convergence for the series The radius R = 1.

  6. R = Check at endpoints to conclude that The interval of convergence is (2,8). Example Find the interval of convergence and the radius of convergence for the series

  7. Theorem If the power series has radius of convergence R > 0, then the function defined by is differentiable and continuous on the interval (c – R, c + R) and we have

  8. Examples Find the interval of convergence and the radius of convergence for the following series. The Bessel Function solves Kepler’s equation for planetary motion. The Bessel Function of order 1 can be represented as a power function and is given above.

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