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)

Quote:

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Originally Posted by transgalactic

1-n^2x^2=0
x=+- sqrt(1/n)

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Should be .

Quote:

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Originally Posted by transgalactic

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

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That's one way.

what are the other ways

i see only the extereme point method

Quote:

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Originally Posted by transgalactic

what are the other ways

i see only the extereme point method

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To prove that converges uniformly to 0 we note that for all x, as . In this case, there is no need to find exact extrema of .