I've been asked to prove that the following integral diverges∫∞0|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 n∈Z
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 n≤x<n+1, we have ∫n+1n|sinπxx|dx>1n+1∫n+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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