Re: point convergence and uniform convergence definition question

f'(x)=n(1+n^2x^2)-nx(2n^2x)/dominator squared

f'(x)=0

n(1+n^2x^2)-nx(2n^2x)=0

n+n^3x^2-2x^2n^3=0

n-n^3x^2=0

1-n^2x^2=0

x=+- sqrt(1/n)

f(1/n)=1/2 got it :)

so i see that we take the extreme points of the expression and do a limit on them

correct?(could you please help me with the question bellow about the cauchy serieses,you have been very helpfull)

Re: point convergence and uniform convergence definition question

Quote:

Originally Posted by

**transgalactic** 1-n^2x^2=0

x=+- sqrt(1/n)

Should be $\displaystyle x = \pm1/n$.

Quote:

Originally Posted by

**transgalactic** so i see that we take the extreme points of the expression and do a limit on them

correct?

That's one way.

Re: point convergence and uniform convergence definition question

what are the other ways

i see only the extereme point method

Re: point convergence and uniform convergence definition question

Quote:

Originally Posted by

**transgalactic** what are the other ways

i see only the extereme point method

To prove that $\displaystyle f_n(x)=(\sin x+\cos x)/n$ converges uniformly to 0 we note that for all x, $\displaystyle |(\sin x+\cos x) / n-0|\le2/ n\to0$ as $\displaystyle n\to\infty$. In this case, there is no need to find exact extrema of $\displaystyle f_n(x)$.