MATA37H3 Lecture Notes - Lecture 21: Limit Comparison Test, Ibm System P, Integral Test For Convergence
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Mata37 - lecture 21 - p-series test, limit comparison test, and alternating series test. 1 np diverges np , p r+: For a p-series with p = 1 1, by p-series test, 1 n diverges: recall integral test: if f (x) is positive, strictly decreasing, continuous on [1, ), such that an = f (n) n n, then an converges f (x)dx converges. (cid:88) (cid:90) : suppose the following: n=1. 1: f (x) is positive on [1, , f (x) is decreasing on [1, , f (x) is continuous on [1, , want to show an converges f (x)dx converges f (x)dx + . exists, because of #3 and n=1. A f (x)dx + + (cid:90) 3 f (x)dx + 2 n f (x)dx converges (exists) (cid:90) . 1 (cid:90) n+1 n f (x)dx < f (n)(n + 1 n), by integral inequality f (x)dx < f (n), i. e. an+1 < bn < an.
