Math Help - point convergence and uniform convergence definition question

1. 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

2. Re: point convergence and uniform convergence definition question

Originally Posted by transgalactic
1-n^2x^2=0
x=+- sqrt(1/n)
Should be $x = \pm1/n$.

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.

3. Re: point convergence and uniform convergence definition question

what are the other ways

i see only the extereme point method

4. Re: point convergence and uniform convergence definition question

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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