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Math Help - Epsilon Convergence Proof

  1. #1
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    Epsilon Convergence Proof

    Prove that the sequence sum(i=1 --> n) 1/(10^i) converges.

    I'm pretty sure it converges to 1/9 but how to we get there by fixing epsilon?
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  2. #2
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    I'm not sure what you mean by "epsilon convergence" but this is a geometric series.

    You should know that for a finite geometric series

    S_n = \frac{a(1 - r^n)}{1 - r}.

    Therefore it is convergent.


    You should also be able to see that if you make n \to \infinity, i.e. make this an infinite series, since |r| < 1, that means r^n \to 0 and thus the sum goes to S_{\infty} = \frac{a}{1 - r}, which is also convergent.
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  3. #3
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    We need to prove convergence by fixing ε > 0 and showing absval(a_n - L) = absval((sum(i=1 ->n) 1/(10^i) - 1/9) < ε
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  4. #4
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    1) You can simply write | | instead of absval, even with LaTeX.
    2) As Prove It said,
    \displaystyle \lim_{n \to \infty} \sum_{i=1}^n \frac{1}{10^i} = \lim_{n \to \infty} \frac{1}{10} \cdot \frac{1 - 10^{-n}}{1 - 10^{-1}}

    Double click on the latex to see the correct code.
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