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I am trying to solve the following problems. Do you have any hints?

Let $g_n(x)$ be functions defined on interval $I = [a,b]$ and suppose $g(x) =\lim_{n \to \infty} g_n(x)$ is defined for every $x \in I$.

  1. $(a)$ If every $g_n(x)$ is continuous, does it follow that $g(x)$ is continuous?
    $(b)$ If that convergence is uniform in $I$, is the limit function $g$ is continuous?

  2. If the convergence is uniform, does it follow that the limit of the integral of $g_n(x)$ from $a$ to $b$ is the integral of the limit?

  3. If $\sum_{n \geq 0} c_n X^n$ has radius of convergence $\rho >0$, then $g(z) = \sum_{\geq 0}c_n z^n$ is defined in the disk $D_\rho = \{z: |z|\lt p\}$ / Give an example that does not converge uniformly in $D_\rho$.

  4. If $0\lt r\lt\rho$, then this series converges uniformly in $D_r$.

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    Wikipedia has all the answers. See [here](http://en.wikipedia.org/wiki/Uniform_convergence) and [here](http://en.wikipedia.org/wiki/Radius_of_convergence)2011-04-27
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    Hello George. I see that you are a new user here. In the future when you ask a question, try to tell us what you have tried so far, and what you are struggling with. That makes it a lot easier to answer, since most people here won't solve your homework question for you. Also, writing only the homework question without any other explanation can be interpreted as a bit rude. I have edited things slightly, but _please_ tell us what you have tried, and what you have done so far, otherwise it is hard to help.2011-04-27
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    Apologies. I wasn't sure how this site is set up to be honest. This is actually not a homework problem, but the first question on a review sheet for a midterm. I couldn't figure it out, panicked, and posted on here. I'll be sure to provide more information as to what I've attempted next time.2011-04-28

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