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11.2 P ROPERTIES OF P OWER S ERIES Math 6B Calculus II

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P OWER S ERIES Power Series centered at x = 0. Power Series centered at x = a.

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T HE F UNCTION 1/(1 – X ) The function 1/(1 – x ) can be rewritten into a power series.

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H OW TO T EST A P OWER S ERIES FOR C ONVERGENCE 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 < x < 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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H OW TO T EST A P OWER S ERIES FOR C ONVERGENCE If the interval of absolute convergence is a – R < x < a + R, the series diverges for

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T HE R ADIUS AND I NTERVAL OF C ONVERGENCE 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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T HE R ADIUS AND I NTERVAL OF C ONVERGENCE The series converges absolutely for every x The series converges at x = a and diverges everywhere else. ( R = 0 )

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C OMBINING P OWER S ERIES

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D IFFERENTIATION AND I NTEGRATION OF A P OWER S ERIES

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The radii of convergence of the power series in Equations (i) and (ii) are both R.

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