Uniformly convergent sequence

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Main Question or Discussion Point

I need to determine whether the sequence [tex]\{\frac{n^2x}{1+n^3x}\}[/tex] is uniformly convergent on the intervals:

[1,2]
[a,inf), a>0

For the first one, I notoced the function is decreasing on the interval, so the [tex]\sup|\frac{n^2x}{1+n^3x}|[/tex] will be when x=1, and when x=1, the sequence goes to 0, proving uniform convergence.

I'm not so sure how to approach the second one, because the sequence may not necessarily be decreasing on [a,inf)

Any help?
 

Answers and Replies

mathman
Science Advisor
7,714
398
The general idea is that for a>0, there will be an N such that the sup of the sequence will be at x=a for n>N.
 

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