Define f:[0,1]→R by
f(x)=∞∑n=1x2(1−x2)n−1
Check that f is continuous or not.
Attempt: f(x)=∞∑n=1x2(1−x2)n−1=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.
No comments:
Post a Comment