MATA37H3 Lecture Notes - Lecture 20: Integral Test For Convergence, Ibm System P, Improper Integral
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Mata37 - lecture 20 - integral test and p-series. Integral test: see page 619, if f (x) is positive, continuous, decreasing on [1, ), and f (n) = an n n, then (cid:90) f (x)dx converges, i. e. both either converge or diverge. (cid:88) an converges n=1: example 1: does. F is decreasing on [3, ); moreover, f is differentiable on [3, ), and so f is continu- ous on [3, ), because diff = cont . * f(cid:48)(x) = (x2 + 1) 12x = (cid:90) a (cid:90) . By integral test, our series converges f (x)dx converges. P-series: see page 619, a series of the form, examples: 1 np , where p r+ (i. e. p r, p > 0) is called a p-series. 1 nn , (not a p-series; p must be xed value: example 2: for what p values does. F (x) is decreasing on [1, ).
