2. LIMIT COMPARISON TEST: SPECIAL CASES The Limit Comparison Test is most often used when the limit L= lim ak lies in the interval (0, oo). In the case where L is zero or infinity, we can still compare a, and be, but only with additional assumptions. Theorem. [Limit Comparison Test: 0- cases] Suppose that ak>0 and b>0 for all k2N, where N is a positive integer. (1) [0 CASE] IF lim ak +00 by (2) [cc-CASE] IF lim k-00 = 0 AND E b, converges THEN a, converges. = ~ AND 2 & diverges THEN ak diverge x (2) Use results from the PreLab and the 0-case of the Limit Comparison Test to prove the convergence of 2) Use results from the PreLab and theco-case of the Limit Comparison Test to prove the divergence of In k k IM8 (Use the 0-case of the Limit Comparison Test to prove the convergence of In k le=1 (D) Use the co-case of the Limit Comparison Test to prove the divergence of 8 WI In k k0.9