MATA37H3 Lecture Notes - Lecture 19: Direct Comparison Test, Contraposition
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Mata37 - lecture 19 - properties of convergant series. 3n2 5 diverges by divergence test: if, see page 618 lim n an (cid:54)= 0, then(cid:88) < 1, this series converges by geometric series test. By property #1, our original series convergant = convergant n=0. Comparison test for series: see page 626, let(cid:88) bn be series, convergance case: if. 0 an bn n n, and bn converges an also converges: divergance case: if. 0 bn an n n, and bn diverges an also diverges. Suppose 0 an bn n n (#1), and bn = t r converges (#2) (cid:88) an converges, i. e. want to show n sn exists lim. Let sn = a1 + a2 + + an, tn = b1 + b2 + + bn. = for each a n, sn is bounded above (by t) Moreover, observe {sn} is increasing, by #1.
