Analysis, converging functions

Let f_n(x)=nx^n for 0<=x<=1 and f_n(x)=0 elsewhere.  Does f_n converge, and if so, to what function? 

It 'converges' to f(x)= inf at 1, 0 elsewhere. So, no, it doesn't, actually.

>>2
How would one go about showing that f(x)=0 for 0<=x<1? 

the power series x^n has radius of convergence (-1,1).

when n gets big the power series converges to zero.

it converges pointwise to f(x) = infinity at 1
                               = 0 elsewhere


It does not converges uniformly to this though, so it does not converges.

This can be seen obviously as for any n, there exists an x < 1 such that nx^n = 1/2   ie  nlnx = -ln2n

so x = exp(-1/n(ln(2n))


I'm not bothering to write out the definition for uniform convergence of sequences of functions, but if you can't see why that contradicts it, you're going to fail analysis anyway