Let f : [a , b] ( ( an integrable function. Prove that:
lim n-> plus infinity S f(x) cos (nx) dx = 0
lim n-> plus infinity S f(x) sin (nx) dx = 0
where S is the integral going from a to b.
Mathematics Homework Solutions
#3312
Real Analysis Problem
We have just finished up integration and are done with a first course in analysis, so chapters 1-6 of Rudin. We are also using the Ross and Morrey/Protter book. Please answer question fully and clearly explaining every step. Any solution short of perfect is useless to me. So if you are not 100% sure whether your answer is right, then please do not answer. The same problem is also attached as a word document with all the symbols. ****************************************************** Let f: [a,b] --> R an integrable function. Prove that:
i) lim *S* f(x) cos(nx) dx = 0
and
ii) lim *S* f(x) sin(nx) dx = 0.
where lim is n as it approaches plus infinity (it is not specified so I believe the default when only n is listed is to plus infinity, I may be mistaken), *S* is my notation for the integral taken from a to b.
This is a proof regarding limits of integrable functions.
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