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∫10x2(1−x2)n−1
Now, Putting x=sinθ, then dx=cosθdθ, therefore integral reduces to
lim∫π/20sin2θ(1−sin2θ)n−1cosθdθ=lim∫π/20sin2θcosnθdθ
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