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Math Help - Series Convergences

  1. #1
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    Series Convergences

    Its a normal summation for K=0 and goes to infinite for  [(-1)^k]/(2k+1)

    how do I show this converges... I know it does but I forgot how, using limit comparison, comparison, divergence, ratio, root, abs. ratio, abs. root, or AST.

    thanks again
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by ryukolink View Post
    Its a normal summation for K=0 and goes to infinite for  [(-1)^k]/(2k+1)

    how do I show this converges... I know it does but I forgot how, using limit comparison, comparison, divergence, ratio, root, abs. ratio, abs. root, or AST.

    thanks again
    Use the Alternatig Series Test using the fact that if a_n=\frac{1}{2n+1} then a_{n+1}<a_n and \lim_{n\to\infty}a_n=0

    Note: \sum_{n=0}^{\infty}\frac{(-1)^n}{2n+1}=\frac{\pi}{4}
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  3. #3
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    so Im digging through my backpack and I find a paper explaining the AST for oscillating harmonics lol

    thanks though..

    Now tell me this... is a multiple of a harmonic only when its 1/k multiplied by a number > 1 ?

    so like 1/2k this would not be a multiple of a harmonic?

    because by AST, we also take the lim as x approaches infinte... 1/2k+1 compares to 1/k .. 1/2k+1 < 1/k aww which proves nothing.. darn it but the limit comp. text proves its 1/2 . Thus the two act similiar so it diverges.. or did I mess that up somewhere?
    Last edited by mr fantastic; December 13th 2008 at 07:55 PM. Reason: Deleted potentially offensive undertones and mild coarse language
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by ryukolink View Post
    so Im digging through my backpack and I find a paper explaining the AST for oscillating harmonics lol
    That is always how it goes
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  5. #5
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by ryukolink View Post
    so Im digging through my backpack and I find a paper explaining the AST for oscillating harmonics lol

    thanks though.. you truly are a math stud.. (no homo)

    Now tell me this... is a multiple of a harmonic only when its 1/k multiplied by a number > 1 ?

    so like 1/2k this would not be a multiple of a harmonic?

    because by AST, we also take the lim as x approaches infinte... 1/2k+1 compares to 1/k .. 1/2k+1 < 1/k aww which proves nothing.. dammit but the limit comp. text proves its 1/2 . Thus the two act similiar so it diverges.. or did I mess that up somewhere?
    No if you have \sum \frac{a}{n} where a is any real constant then your series converge. But I think you are confusing the AST, it states roughly: If you have a series \sum (-1)^na_n and a_{n+1}\leqslant a_{n} and \lim a_n=0 then the series converges.
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  6. #6
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    Quote Originally Posted by ryukolink View Post
    so Im digging through my backpack and I find a paper explaining the AST for oscillating harmonics lol

    thanks though..

    Now tell me this... is a multiple of a harmonic only when its 1/k multiplied by a number > 1 ? Mr F says: This will not define an alternating series ....

    so like 1/2k this would not be a multiple of a harmonic?

    because by AST, we also take the lim as x approaches infinte... 1/2k+1 compares to 1/k .. 1/2k+1 < 1/k aww which proves nothing.. darn it but the limit comp. text proves its 1/2 . Thus the two act similiar so it diverges.. or did I mess that up somewhere?
    ..
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