I have no idea how to prove this. Can someone help?

Given a real number x, denote by [x] the largest integer less than or equal to x. Prove that the function defined

= is Riemann integrable on any interval [a,b]. Then compute

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- May 15th 2008, 01:03 PMnamelessguyRiemann Integrability
I have no idea how to prove this. Can someone help?

Given a real number x, denote by [x] the largest integer less than or equal to x. Prove that the function defined

= is Riemann integrable on any interval [a,b]. Then compute

- May 15th 2008, 03:11 PMNonCommAlg
for n = 0 the term in your series is 0. so i'll assume that n > 0. let

each is integrable on any interval [a, b], because it's continuous almost everywhere on the

interval. let so is integrable on [a, b] for all n.

since by Weierstrass test, is uniformly convergent. so

is integrable on [a, b] and if a = 0 and b = 1, then this gives us:

- May 15th 2008, 03:22 PMThePerfectHacker
I will do the special case . Define a sequence of functions as . Notice that . Since it follows by the Weierstrass test that the series of functions converges uniformly to a function . Each function is discontinous at for . This implies that is integrable and . It remains to compute . The graph of consists of a congruent right-angles triangles each having height and with base width . Thus, . I might have missed maybe on the exponent, but you get the idea. Now this is a geometric series which sums easily.

EDIT: Got beaten in the response by a fellow algebraist. (Hi)