MATH 1272 Lecture Notes - Lecture 1: Ibm System P
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For today, assume f is continuous, decreasing, and positive on [a, ) Why can"t we use riemann sums to estimate. Now that we have series, we could look at using infinitely many rectangles of width x=1. Total area of these rectangles (as an sum) If the improper diverges, so does the series. Now try the right hand rule (keep x =1) Suppose f is a continuous, positive, decreasing function on [b, ) , b 0, natural, and an=f(n) is a sequence. F is positive, since x 0 always. F is decreasing, since f"(x) = -1/x <0 on [1, ) F is positive on [2, ) since x, ln(x) are. F is continuous on [2, ) since x, ln(x)-3 continuous and xln(x) 0. Suppose f is a continuous, positive, decreasing function on [n, ), some natural number. If rn:= s-sn is the nth remainder term, then. Also find upper and lower bounds on s (actual sum)
