Thread: Finding limit as n approaches infinity

1. Finding limit as n approaches infinity

I am not able to prove it by the hint shown. I perfectly understand the question and it is obvious that the antiderivative approaches 0 as n approaches infinity. But the hint about riemann integral is confusing me.

2. Originally Posted by xyz

I am not able to prove it by the hint shown. I perfectly understand the question and it is obvious that the antiderivative approaches 0 as n approaches infinity. But the hint about riemann integral is confusing me.
The idea is that you want to prove that if $\displaystyle f_n=\frac{1}{(1+x^2)^n}$, then $\displaystyle f_n\to0$ uniformly on $\displaystyle [1,2]$. Then you know that you can interchange the limits, i.e:

$\displaystyle \lim_{n\to\infty}\int_1^2(1+x^2)^{-n}\,dx=\int_1^2\left(\lim_{n\to\infty}(1+x^2)^{-n}\right)\,dx=\int_1^2 0\,dx=0$

So all you need to prove is uniform convergence, and then the above equality will hold. (Hint: Use Dini's Theorem.)