Saturday, 27 October 2018

sequences and series - Check that f(x)=suminftyn=1x2(1x2)n1 is continuous or not.

Define f:[0,1]R by



f(x)=n=1x2(1x2)n1



Check that f is continuous or not.



Attempt: f(x)=n=1x2(1x2)n1=lim
Now, Putting x=\sin\theta, then \mathrm dx=\cos\theta\mathrm d\theta, therefore integral reduces to




\lim\int_{0}^{\pi/2}\sin^2\theta(1-\sin^2\theta)^{n-1}\cos\theta\mathrm d\theta\\ =\lim\int_{0}^{\pi/2}\sin^2\theta\cos^n\theta\mathrm d\theta



Now from here the result will depend on n i.e. n=2m,n=2m+1
In these two cases the result will be different, Hence f is not continuous.



am I right? Different approaches are invited. Thank you.

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