Suppose f(a, b] --> R is continuous, monotonically decreasing, goes to oo as x --> a+ but its improper Riemann integral exists over (a, b]. Like f(x) = 1/sqrt(x) over (0, 1].
Let S_n be a sequence of Riemann sums taken over [a, b], associated with partitions P_n, in such a way that: (1) The norm of P_n (length of the largest interval) --> 0 and (2) For each interval of P_n, the tag point, at which f is evaluated, is its the interval right end point.
Show that S_n --> Int a^b f(x) dx
Are these conditions too rigid? Can any of them be dropped without compromising the conclusion?
Thank you
2007-12-18
03:53:42
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3 answers
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asked by
Laura
1
in
Mathematics