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Limits and series

  1. Mar 31, 2005 #1
    First, a quick question about limits. Is it true that if a function "flip-flops" between positive and negative values, the limit does not exist? Say in the case of

    [tex]\lim_{n\rightarrow\infty} (-10)^{n} [/tex] ?

    Second, I'm having some difficulty with a problem and I'm not quite sure how to start.

    )If the nth partial sum of a series [itex]\Sigma_{n=1}^{\infty} a_n [/itex] is [itex] s_n = 3-2^{-n}n [/itex], find [itex]a_n[/itex] and [itex]\Sigma_{n=1}^{\infty} a_n [/itex].
     
  2. jcsd
  3. Mar 31, 2005 #2

    Zurtex

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    To the first problem, this sequence has no limit but this is not always the case for oscillating sequences, for example:

    [tex]\lim_{n \rightarrow \infty} (-10)^{-n} = 0[/tex]

    As for the second problem, have you tried calculating [itex]S_n - S_{n-1}[/itex]?
     
  4. Mar 31, 2005 #3

    NateTG

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    No. Although the limit that you list does not exist. An example of a limit that 'flip-flops' but does exist would be
    [tex]\lim_{n\rightarrow\infty} \left(\frac{-1}{10}\right)^{n}[/tex]


    Can you find the first term of the series?
    Once you have that, can you find the second?

    Also
    [tex]\sum_{n=1}^{\infty} a_n = \lim_{n \rightarrow \infty} s_n[/tex]
    by definition.

    P.S. This is not all that important, but if you use /sum instead of /sigma then LaTeX will automatically place the sub and superscipts in the right places.
     
  5. Mar 31, 2005 #4

    dextercioby

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    No.In inline text ("itex" tags),it won't.[itex]\sum_{k=1}^{\infty} [/itex]...:wink:

    Also,not to get confused,u might use "k" as a dummy summation index.It'd be [itex]\sum_{k=1}^{n} a_{k} [/itex]...

    Daniel.
     
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