Thursday, 14 February 2013

calculus - How to prove absolute summability of sinc function?



We know that 0(sinxx)2dx=0sinxxdx=π2.



How do I show that 0|sinxx|dx

converges?


Answer



It doesn't. Using the convexity of 1/x,




0|sinxx|dx=k=0(k+1)πkπ|sinxx|dx>k=0(k+1)πkπ|sinx|(k+1/2)πdx=2πk=01k+1/2,



which diverges since the harmonic series diverges.


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