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Test the following for uniform convergence:

(a) The sequence of functions {$x^n/(1 + x^n)$} over the interval [0, 2].
(b) The series $ ∑_{n=1}^∞\sin nx/(n^2+ 1)$over $\mathbb{R}$.
(c) The sequence of functions{$n^2x^2e^{−nx}$} over the interval (0, ∞).

How can I solve this problem?

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What have you tried? – emka Jan 14 at 17:23
Which one - or you want generic hints common to all three? – Jan Dvorak Jan 14 at 17:32
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(a) Calculate the pointwise limit and think about the result while keeping in mind that a uniform limit of continuous functions must be continuous. (b) Use the Weierstrass $M$-test. (c) Calculate the pointwise limit, and calculate $f_n(1/n)$, where $f_n(x)=n^2x^2e^{-nx}$. Think about the results... – David Mitra Jan 14 at 17:52

1 Answer

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These are just short, potentially useful observations.

(a) Notice that $\frac{x^n}{x^n+1} \leq 1$ for $x \in [0,1]$

(b) $\sum\frac{\sin(nx)}{n^2+1}\leq \sum \frac{1}{n^2+1}\leq \sum \frac{1}{n^2}...$

(c) This converges uniformly, but I have to think about it more.

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For c), note $f_n(x)=n^2x^2e^{-nx}$ has value $1/e$ for $x=1/n$; while the pointwise limit of $(f_n)$ is the zero function on $(0,\infty)$. – David Mitra Jan 14 at 17:44
Thanks. As I noted, I don't know if my answer is particularly useful. I've had these problems before and the information I provided for (a) and (b) helped me think about convergence and uniform convergence. – emka Jan 14 at 21:32

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