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Thread: converging interval for power series

  1. #1
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    converging interval for power series

    is it possible to have a powerr series that converges with interval of 0 inclusive to inf?
    Last edited by lc99; Apr 4th 2018 at 07:52 PM.
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  2. #2
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    Re: converging interval for power series

    $e^x = \sum \limits_{n=0}^\infty \frac1{n!}x^n$ converges everywhere.

    Of course, you can't include "infinity" in the interval of convergence.
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    Re: converging interval for power series

    the Taylor series of any function bounded on that interval converges everywhere on it.

    for example

    $\cos(x) = \sum \limits_{k=0}^\infty~\dfrac{(-1)^n x^{2n}}{2n!}$

    converges for all $x \in \mathbb{R}$
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    Re: converging interval for power series

    is there a reason why we cant have infinity like [0,inf)?
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    Re: converging interval for power series

    Quote Originally Posted by lc99 View Post
    is there a reason why we cant have infinity like [0,inf)?
    You want it to only converge on that interval?

    I'm not understanding what you are asking.
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    Re: converging interval for power series

    Quote Originally Posted by lc99 View Post
    is there a reason why we cant have infinity like [0,inf)?
    No, there isn't. In his first response Archie was concerned that you "inclusive" referred to the "inf" as well as to 0. Yes, you can include 0 in an interval, no you cannot include "infinity" because that is not a number in the sense being used here. You can have [0, inf). You cannot have [0, inf].
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  7. #7
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    Re: converging interval for power series

    for a power series we cannot have an interval of convergence of the form $\displaystyle [0,\infty )$

    the reason is that if a power series converges at $\displaystyle x=a\neq 0$

    then it also converges for all $\displaystyle x$ with $\displaystyle |x|<|a|$
    Last edited by Idea; Apr 5th 2018 at 06:20 AM.
    Thanks from topsquark and Archie
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