Be the function $$f(x) = \ \int_{\frac{\pi}{2}}^{x} \frac{\cos(t)}{t - \frac{\pi}{2}}dt.$$ I tried to find the Taylor's series of $f(x)$, but I didn't succeed. What is the Taylor series of $f(x)$? Is there a clever way to find it?

Please let me know if the question is unclear.


Hint: $\cos t=\sin(\pi/2-t)$. Change the variable and then integrate the Maclaurin's formula for $\frac{\sin x}{x}$.

  • $\begingroup$ Could you develop your answer? $\endgroup$ – J.Doe Oct 13 '17 at 14:14
  • $\begingroup$ There are 3 steps: (1) variable change in the integral, (2) getting Maclaurin for the resulting integrand and (3) integration. When you try going, which step do you get stuck at? $\endgroup$ – A.Γ. Oct 13 '17 at 14:23
  • $\begingroup$ Why could we integrate an infinite sum? I mean each term of the infinite sum? It is clear we could do it by linearity of a finite sum, but why on an infinite sum? $\endgroup$ – J.Doe Oct 13 '17 at 14:25
  • $\begingroup$ @J.Doe Power series can be differentiated and integrated within their radius of convergence (because the convergence is uniform there). Here the radius is infinity (because of $\sin$). $\endgroup$ – A.Γ. Oct 13 '17 at 14:42

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