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Math Help - analysis 4.1.1

  1. #1
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    analysis 4.1.1

    4.1.1

    How many terms of the series
    1 + 1/2 + 1/4 + 1/8 + ...
    do we need to take if we are to guarantee that we are within є = 0.0001 of the target value 2?
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  2. #2
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    Quote Originally Posted by pila0688 View Post
    How many terms of the series
    1 + 1/2 + 1/4 + 1/8 + ...
    do we need to take if we are to guarantee that we are within є = 0.0001 of the target value 2?
    Let a=\tfrac{1}{2} then 1+a+...+a^n = \frac{1-a^n}{1-a} = \frac{1}{1-a} - \frac{a^n}{1-a}

    Therefore, \left| \sum_{k=0}^n a^k - 2 \right| \leq <br />
\left(\frac{1}{2}\right)^{n-1}

    Now make the error term sufficiently small.
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