Monday, January 1, 2018

calculus - Prove that the integral intinfty0bigg|fracsin(pix)xbigg|dx diverges



I've been asked to prove that the following integral diverges0|sin(πx)x|dx
The thing is, that the textbook explicitly suggests a way to prove that. It states the following:




Start by showing that the integral b0|sin(πx)x|dx is bounded below by n0|sin(πx)x|dx with a respective integer nZ





I can't figure out what they mean by that - how can a function be bounded below by itself? It would be great if someone could clear that up for me.


Answer



Hint: For nx<n+1, we have n+1n|sinπxx|dx>1n+1n+1n|sinπx|dx=1n+1|n+1nsinπxdx| This follows just from properties of fractions; the integral that remains is easy to calculate. Now use the fact that 1n diverges.


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