# point convergence and uniform convergence definition question

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• Apr 12th 2012, 12:45 PM
transgalactic
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
• Apr 12th 2012, 01:15 PM
emakarov
Re: point convergence and uniform convergence definition question
Quote:

Originally Posted by transgalactic
1-n^2x^2=0
x=+- sqrt(1/n)

Should be $x = \pm1/n$.

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.
• Apr 12th 2012, 01:18 PM
transgalactic
Re: point convergence and uniform convergence definition question
what are the other ways

i see only the extereme point method
• Apr 12th 2012, 02:34 PM
emakarov
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 $f_n(x)=(\sin x+\cos x)/n$ converges uniformly to 0 we note that for all x, $|(\sin x+\cos x) / n-0|\le2/ n\to0$ as $n\to\infty$. In this case, there is no need to find exact extrema of $f_n(x)$.
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