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VeeEight
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Here is my problem that seems to be easy but I cannot seem to manage to find the answer
let f(x) = 1 if x=1/n for some positive integer n
= 0 if else
the question asks to show f Riemann integrable and find the integral of this function over [0,1]
f on [a,b] is Riemann integrable if U(f) = L(f) (where U(f) is the inf of the set of upper sums of f with respect to p in P, where p is a partition of [a,b] and P is the set of all partitions of [a,b]. And L(f) is defined similarly replacing inf with sup and upper with lower)
To show that it is Riemann integrable I construct a sequence of partitions of [0,1] where the nth partition is defined as [0,1] being split up into n (equal sized) pieces. so x_k = k/n for some n in the naturals. Then I showed that U(f,P_n) - L(f,_n) = 1/n and so it's limit is 0. Thus by the sequential criterion for integrability (I believe it's called Cauchy criterion for Riemann integrability), f is integrable on [0,1]. I think this argument is correct.
I am having trouble evaluating the integral and request some help. I know the standard techniques for integrals (parts, substitution, etc) but i can't seem to find a use for them, which makes me think I should use a theorem to reduce the integral to a simple case and then the result will follow. We just started integration so the theorems we know are about continuous functions, functions with a finite (or content zero) number of discontinuities, and a few corollaries to the previous theorems.
Homework Statement
let f(x) = 1 if x=1/n for some positive integer n
= 0 if else
the question asks to show f Riemann integrable and find the integral of this function over [0,1]
Homework Equations
f on [a,b] is Riemann integrable if U(f) = L(f) (where U(f) is the inf of the set of upper sums of f with respect to p in P, where p is a partition of [a,b] and P is the set of all partitions of [a,b]. And L(f) is defined similarly replacing inf with sup and upper with lower)
The Attempt at a Solution
To show that it is Riemann integrable I construct a sequence of partitions of [0,1] where the nth partition is defined as [0,1] being split up into n (equal sized) pieces. so x_k = k/n for some n in the naturals. Then I showed that U(f,P_n) - L(f,_n) = 1/n and so it's limit is 0. Thus by the sequential criterion for integrability (I believe it's called Cauchy criterion for Riemann integrability), f is integrable on [0,1]. I think this argument is correct.
I am having trouble evaluating the integral and request some help. I know the standard techniques for integrals (parts, substitution, etc) but i can't seem to find a use for them, which makes me think I should use a theorem to reduce the integral to a simple case and then the result will follow. We just started integration so the theorems we know are about continuous functions, functions with a finite (or content zero) number of discontinuities, and a few corollaries to the previous theorems.