AP CALCULUS BC Section 9.2: SERIES AND CONVERGENCE, pg. 606 INFINITE SERIES If an is an infinite sequence, the aaaaann 1234... a ... is an infinite series. (For n1 some series is convenient to begin the index at n=0) To find the sum of an infinite series, consider the following sequence of partial sums. Sa11 Saa212 Saaa3123 ... Saaann123... a If this sequence of partial sums converges, the series is said to converge and has the sum indicated in the following definition. AP CALCULUS BC Section 9.2: SERIES AND CONVERGENCE, pg. 606 Sample problem #1: CONVERGENT AND DIVERGENT SERIES Determine if the given series is convergent or divergent. 3 a) 3 n2 52n AP CALCULUS BC Section 9.2: SERIES AND CONVERGENCE, pg. 606 n 4 b) n1 3 AP CALCULUS BC Section 9.2: SERIES AND CONVERGENCE, pg. 606 11 c) n1 nn1 The series in (c) is a telescoping series of the form bb12 bb 23 bb 34 bb 45 ... Note that all b(s), from b2 to bn , are canceled, therefore in a telescoping series Sbbnn 11 It follows that a telescoping series will converge if and only if bn approaches a finite number as n . Moreover, if the series converges, its sum is Sb 11 lim bn n AP CALCULUS BC Section 9.2: SERIES AND CONVERGENCE, pg. 606 Sample problem #2: WRITING SERIES IN TELESCOPING FORM 3 Determine the sum of the series 2 n1 12n 2 AP CALCULUS BC Section 9.2: SERIES AND CONVERGENCE, pg. 606 GEOMETRIC SERIES A geometric series has the form arnn a ara r 2 ... ar .. a 0 (ratio: r) n0 Sample problem #3: CONVERGENT AND DIVERGENT GEOMETRIC SERIES Determine, if possible, the sum of the given series. 5 5 a) b) n n n2 8 n0 3 AP CALCULUS BC Section 9.2: SERIES AND CONVERGENCE, pg. 606 c) A GEOMETRIC SERIES FOR A REPEATING DECIMAL Use a geometric series to write 0.05 as the ratio of two integers. AP CALCULUS BC Section 9.2: SERIES AND CONVERGENCE, pg. 606 AP CALCULUS BC Section 9.2: SERIES AND CONVERGENCE, pg. 606 Nth –TERM TEST FOR DIVERGENCE The following theorem states that if a series converges, the limit of its nth term must be 0. The contrapositive of the above theorem provides a useful test for divergence. This nth-Term for Divergence states that is the limit of the nth term of a series does not converge to 0, the series must diverge. AP CALCULUS BC Section 9.2: SERIES AND CONVERGENCE, pg. 606 Sample problem #3: USING THE nth-TERM TEST FOR DIVERGENCE Determine, if possible, if the series diverges based on the nth-Term for Divergence. n n 2!n 1 a) 5 b) c) n0 n0 13! n n0 5 .