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Math Help - Inductive sequences

  1. #1
    Junior Member utopiaNow's Avatar
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    Inductive sequences[Unsolved]

    Inductively define a sequence \{x_n\} by x_1 = 1 and x_n = x_{n-1} - \frac{(-1)^n}{n}. Show that \{x_n\} converges.

    I'm having trouble proving this, my idea was to show that this sequence was Cauchy and hence convergent(since we're in \mathbb{R}). However I'm having difficulty showing its Cauchy as for arbitrary \epsilon > 0, I need |x_m - x_n| < \epsilon where this holds for some integer N, and \forall m, n \geq N. My problem is with how to choose an N such that |x_{m-1} - \frac{(-1)^m}{m} - x_{n-1} + \frac{(-1)^n}{n}| < \epsilon holds for arbitrary \epsilon > 0.

    Also another similar question:
    Inductively define a sequence \{s_n\} by s_1 = 3 and s_{n+1} = \frac{2}{3}s_n + \frac{4}{3s_n}. Prove \{s_n\} converges and evaluate the limit.

    My idea to show that it converges was to show that its monotonically decreasing and bounded. And then I know the limit is 2, since I played around with it on my calculator. However I had problems when I tried to use induction to show that its monotonically decreasing, the equations get really messy and I never get anything nice I can use.

    Also a similar things happens when, for abitrary \epsilon > 0 I try find N such that \forall n \geq N, |s_n - 2| < \epsilon. The definition of \{s_n\} is not in terms of n, so I'm not sure how to find an N that meets the requirement.

    Any hints would be greatly appreciated.
    Last edited by utopiaNow; October 25th 2009 at 01:23 PM.
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by utopiaNow View Post
    Inductively define a sequence \{x_n\} by x_1 = 1 and x_n = x_{n-1} - \frac{(-1)}{n}. Show that \{x_n\} converges.

    I'm having trouble proving this, my idea was to show that this sequence was Cauchy and hence convergent(since we're in \mathbb{R}). However I'm having difficulty showing its Cauchy as for arbitrary \epsilon > 0, I need |x_m - x_n| < \epsilon where this holds for some integer N, and \forall m, n \geq N. My problem is with how to choose an N such that |x_{m-1} - \frac{(-1)}{m} - x_{n-1} + \frac{(-1)}{n}| < \epsilon holds for arbitrary \epsilon > 0.
    Do you mean x_{n-1}-\frac{(-1)^{\color{red}n}}{n}?

    If so, look what happens when you write out the first few terms:

    x_1=1
    x_2=1-\frac{1}{2}
    x_3=1-\frac{1}{2}+\frac{1}{3}
    ...

    So x_n will be the nth partial sum of \sum_{k=1}^{\infty}\frac{(-1)^{n+1}}{n}, which converges by the alternating series test (to \ln2, but that's irrelevant for this problem).
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  3. #3
    Junior Member utopiaNow's Avatar
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    Quote Originally Posted by redsoxfan325 View Post
    Do you mean x_{n-1}-\frac{(-1)^{\color{red}n}}{n}?
    Oops, my mistake. Yes I did mean that. I read over what I had typed so many times and still I couldn't see my typo. Sorry about that. I'll fix it up top.
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