Lecture Notes

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====== HW 8 ======
Let's consider a few examples of sequences and series of functions. 

1. Let $f_n(x) = \frac{n + \sin x} {2n + \cos n^2 x}$, show that $f_n$ converges uniformly on $\R$. 

2. Let $f(x) = \sum_{n=1}^\infty a_n x^n $. Show that the series is continuous on $[-1, 1]$ if $\sum_n |a_n| < \infty$. Prove that $\sum_{n=1}^\infty n^{-2} x^n$ is continuous on $[-1, 1]$. 

(In general, if one only know that $\sum_n a_n$ and $\sum_n (-1)^n a_n$ converge, then the result still holds, but is harder to prove. See Ross Thm 26.6)

3. Show that $f(x) = \sum_n x^n$ represent a continuous function on $(-1,1)$, but the convergence is not uniform. 
(Hint: to show that $f(x)$ on $(-1,1)$ is continuous,  you only need to show that for any $0<a<1$, we have uniform convergence on $[-a, a]$. Use Weierstrass M-test. )

Lecture Notes

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