Calc_-_Notes_on_geometric_nth_term_telescoping_2.doc - Geometric Series nth Term Test for Divergence and Telescoping Series Geometric Series Test A

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Geometric Series, nth Term Test for Divergence, and Telescoping Series Geometric Series Test A geometric series is in the form 1 1 1 0 1 or , 0. n n n n a r a r a The geometric series diverges if 1 r . If 1, r the series converges to the sum 1 . 1 a S r Ex . Determine whether the following series converge or diverge. (a) 1 3 2 n n (b) 1 3 2 n n (c) 2 1 1 9 4 n n n (d) 3 1 1 ( 4 ) 5 n n n ________________________________________________________________________________ nth Term Test for Divergence If 0 lim n n a   , then the series 1 n n a diverges. Note : This does NOT say that if 0 lim n n a   , then the series converges. This test can only be used to prove divergence. If 0 lim n n a   , then this test doesn’t tell us anything, and we need to use another test. Ex . Determine whether the following series converge or diverge. (a) 1 2 3 3 5 n n n (b) 1 ! 2 ! 1 n n n (c) 1 3 2 3 n n n _________________________________________________________________________________

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