9 4 Radius of Convergence
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1 9 4 Radius of Convergence In this section we will investigate convergence using several tests. This is important because there are different constraints that govern convergence depending on the type of power series that is under investigation. Ex. 1 Illustrating the importance of convergence Consider the mathematical sentence: For what values of x is this an identity? Support your answer graphically. Graphing helps support the idea of convergence, but it does NOT PROVE convergence or divergence. Feb 12 7:37 PM 1
2 1. The convergence Theorem for Power Series There are three possibilities for to convergence: with respect 1) There is a positive number R such that the series diverges for but converges for The series may or may not converge at either of the endpoints. 2) The series 3) The series The number R and the set of all values x for which the series converges is called the The radius of convergence completely determines the interval of convergence if R is either zero or infinite. For we still need to look at the endpoints of the interval. Feb 12 7:42 PM 2
3 Ex. 2 Find the radius of convergence and the interval of convergence for Feb 12 7:44 PM 3
4 2. nth Term Test (For Divergence) This is the most obvious test for convergence. We look for the n th term to approach zero. Ex.3 Determine if the series diverges. converges or Feb 12 7:45 PM 4
5 COMPARING NONNEGATIVE SERIES Sometimes if we do not know if a series converges or diverges, we can look at a similar, known convergent series. This is used with a series that has nonnegative numbers. 3. Direct Comparison Test Let Ʃa n be a series with NO negative terms. (a) Ʃa n converges if (b) Ʃa n diverges Ex.4 Prove that converges for all real x. TRY Prove that converges. Feb 12 7:49 PM 5
6 4. ABSOLUTE CONVERGENCE If the series Ʃ an of absolute values converges, then Ʃan Absolute convergence implies convergence but the reverse is not necessarily true. Ex. 5 Using Absolute Convergence Show that converges for all x. TRY Show that converges for all x. Feb 12 7:50 PM 6
7 5. RATIO TEST Let Ʃa n be a series with positive terms, and with Then, (i) The series converges if (ii) The series diverges if (iii)the test is inconclusive Complete exploration 1 on page 508. Ex. 6 Finding the Radius of Convergence Find the radius and the interval of convergence for TRY Find the radius and the interval of convergence for Feb 12 7:53 PM 7
8 Ex. 7 Radius of Convergence 0 Find the radius of convergence of the series TRY Find the radius of convergence of the series Feb 12 7:55 PM 8
9 Ex. 8 Determining convergence of a series Determine the convergence or divergence of the series Feb 12 7:56 PM 9
10 6. Telescoping Series A telescoping series is so called because the partial sums all cancel out (collapse) and we are left with the initial and the final terms the two ends of a hand held telescope. Ex.9 Find the sum of Feb 12 7:57 PM 10
11 ENDPOINT CONVERGENCE We need to take into consideration endpoints when investigating convergence. Complete Exploration 2 on page 509 Feb 12 7:57 PM 11
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