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AP Calculus BC – Chapter 9 - Infinite Series 9.4: Radius of Convergence

AP Calculus BC – Chapter 9 - Infinite Series 9.4: Radius of Convergence. Goals : Use the nth-Term Test, the Direct Comparison Test, and the Ratio Test to determine the convergence or divergence of a series of constants or the radius of convergence of a power series.

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AP Calculus BC – Chapter 9 - Infinite Series 9.4: Radius of Convergence

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  1. AP Calculus BC – Chapter 9 - Infinite Series 9.4: Radius of Convergence Goals: Use the nth-Term Test, the Direct Comparison Test, and the Ratio Test to determine the convergence or divergence of a series of constants or the radius of convergence of a power series.

  2. The Convergence Theorem for Power Series: Theorem: There are three possibilities for with respect to convergence: 1. There is a positive number R such that the series diverges for |x – a|>R but converges for |x – a|<R. The series may or may not converge at either of the endpoints x=a – R and x=a + R. 2. The series converges for every x (R = ∞). 3. The series converges at x = a and diverges elsewhere (R = 0).

  3. Radius & Interval of Convergence: The number R is the radius of convergence, and the set of all values of x for which the series converges is the interval of convergence. The radius of convergence completely determines the interval of convergence if R is either zero or infinite. For 0<R<∞, there remains the question of what happens at the endpoints.

  4. The nth-Term Test for Divergence: Theorem: diverges if fails to exist or is different from zero.

  5. The Direct Comparison Test: Theorem: Let ∑an be a series with no negative terms. (a) ∑an converges if there is a convergent series ∑cn with an ≤cn for all n>N, for some integer N. (b) ∑an diverges if there is a divergent series ∑dn of nonnegative terms with an ≥dn for all n>N, for some integer N.

  6. Absolute Convergence: Absolute convergence: If the series ∑|an| of absolute values converges, then ∑anconverges absolutely. Theorem: If ∑|an| converges, then ∑an converges.

  7. The Ratio Test: Theorem: Let ∑an be a series with positive terms, and with Then, (a) the series converges if L<1. (b) the series diverges if L>1. (c) the test is inconclusive if L=1.

  8. Assignments and Note: • CW: Explorations 1 & 2. • HW 9.4: #3, 9, 15, 21, 27, 38, 40. • Test Thursday, March 1.

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