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=lim10x2(1x2)n1


Now, Putting x=sinθ, then dx=cosθdθ, therefore integral reduces to




limπ/20sin2θ(1sin2θ)n1cosθdθ=limπ/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

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find limh0sin(ha)h without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...