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Math Help - Power Series - lim sup proof

  1. #1
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    Power Series - lim sup proof

    Let sum(a_nx^n) be a power series with a_n not zero, and assume L=lim|a_(n+1)/a_n| exists.
    a) Show that if L is not zero, then the series converges for all x in (-1/L,1/L).
    b) Show that if L=0, then the series onverges for all x in R
    c) Show that a) and b) continue to hold if L is replaced by the limit
    L'=lim(s_n) where s_n=sup{|a_(k+1)/a_k|:k>=n}
    The value L' is called the limit superior or lim sup of the sequence |a_(n+1)/a_n|. It exists iff the sequence is bounded.
    d) Show that if |a_(n+1)/a_n| is unbounded, then the original series converges only when x=0

    I'm looking at this and I have no clue how to start.
    Like for a), I start by assuming L is not zero.
    So we have lim|a_(n+1)/a_n|is nonzero. Then I have trouble getting further.
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  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    I assume you know that this is the ratio test!
    Ratio test - Wikipedia, the free encyclopedia
    You can use the link above or any (Advanced) Calculus book.
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  3. #3
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    yeah I see that it's the ratio test, but don't see where the (-1/R,1/R) comes from and I guess I don't see where we use the a_nx^n
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  4. #4
    Super Member girdav's Avatar
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    If you fix an x\in\mathbb R and let b_n :=a_nx^n, what about \displaystyle \lim_{n\to\infty}\left|\frac{b_{n+1}}{b_n}\right|?
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  5. #5
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    Then by ratio test, if L<1 series converges absolutely
    If L>1 series does not converge
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  6. #6
    Super Member girdav's Avatar
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    Quote Originally Posted by mathematic View Post
    Then by ratio test, if L<1 series converges absolutely
    If L>1 series does not converge
    The condition is on x, not on L because it is given by the sequence \left\{a_n\right\}.
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  7. #7
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    I guess I don't understand then, because I thought a) was asking about L
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  8. #8
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    Ok I figured out a) and b)
    Now for c) I need to look at sup. I get that, but am unsure on dealing with. Like I know I use ratio test again.
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  9. #9
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    and sup=smallest upper bound. It's just applying it that has me stuck
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