MAT 21C Lecture Notes - Lecture 7: Alternating Series Test, Limit Comparison Test, Alternating Series
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MAT 21C Full Course Notes
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=(cid:2869: alternating series have the form (cid:4666) (cid:883)(cid:4667)+(cid:2869)(cid:1853)= (cid:1853)(cid:2869) (cid:1853)(cid:2870)+(cid:1853)(cid:2871) (cid:1853)(cid:2872)+ . =(cid:2869: theorem suppose (a) (cid:1853)>(cid:882), (b) (cid:1853)+(cid:2869) (cid:3409) (cid:1853), and (c) lim (cid:1853)=(cid:882). If all three conditions are met, then (cid:4666) (cid:883)(cid:4667)+(cid:2869)(cid:1853) Then (a) ln(cid:4672)(cid:883)+(cid:2869)(cid:4673) > 0 for all n. =(cid:2869) (b) ln(cid:4672)(cid:883)+ (cid:2869)+(cid:2869)(cid:4673)(cid:3409) ln(cid:4672)(cid:883)+(cid:2869)(cid:4673) since (cid:883)+ (cid:2869)+(cid:2869) < (cid:883)+(cid:2869) (cid:3643) (cid:2869)+(cid:2869) < (cid:2869) (cid:3643) n < n + 1 (c) lim ln(cid:4672)(cid:883)+(cid:2869)(cid:4673)=ln (cid:4666)lim (cid:883)+(cid:2869)(cid:4667)=ln(cid:4666)(cid:883)(cid:4667)=(cid:882). If not, then series diverges and if so, then no: geometric: converges if ||(cid:883), diverges if (cid:3409)(cid:883) (cid:2869) =(cid:2869: 5) for alternating series, try the alternating series test, 4) for series with positive and negative terms, check for absolute convergence |(cid:1853)| By the nth term test, the series diverges since. Instead, we can apply the limit comparison converges by the root test. =(cid:2869) converges by the p-series where p = 2 and comparison test (due to absolute value sign). (cid:4667).
