MAT 21C Lecture Notes - Lecture 6: Absolute Convergence, Convergent Series, Ratio Test

• School: UC-Davis
• Department: Mathematics
• Course: MAT 21C
• Professor: John Hunter

Document Summary

Mat 21c lecture 6 absolute convergence, ratio and root tests absolutely converges. absolutely convergent? converges. of positive and negative terms is converges. Absolute convergence of a series implies convergence of a series. does not absolute converge. , in fact, converges to ln(2). (cid:2869) =(cid:2869) which: an important property of absolutely convergent series is that the series, absolutely convergent series: a series . =(cid:2869) converges, then : theorem if || =(cid:2869: example: does the series absolutely converge? (cid:4666) (cid:2869)(cid:4667)+(cid:3117) = 1 (cid:2869)(cid:2870)(cid:3118) + (cid:2869)(cid:2871)(cid:3118) (cid:2869)(cid:2872)(cid:3118) + (cid:4666) (cid:2869)(cid:4667)+(cid:3117) = 1 + (cid:2869)(cid:2870)(cid:3118) + (cid:2869)(cid:2871)(cid:3118) + (cid:2869)(cid:2872)(cid:3118) + this is a p-series with p = 2, which converges. =(cid:2869) (cid:3118: example 2: is the series (cid:4666) (cid:2869)(cid:4667)+(cid:3117) = 1 (cid:2869)(cid:2870) + (cid:2869)(cid:2871) (cid:2869)(cid:2872) + (cid:4666) (cid:2869)(cid:4667)+(cid:3117) = 1 + (cid:2869)(cid:2870) + (cid:2869)(cid:2871) + (cid:2869)(cid:2872) + this represents the harmonic series . =(cid:2869: ratio test: suppose {} is a sequence of nonzero terms in which the lim |+(cid:3117)|