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    A friend asked me this question, and neither of us could see where he went wrong (didn't help that I don't know much about remainder terms in Taylor expansions), so any input from anyone here would be appreciated.

    The expansion
    arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} .... + (-1)^n\frac{x^{2n+1}}{2n+1}...
    can be used to approximate \pi by substituting in x=1, to give
    \pi = 4(1 - \frac{1}{3} + \frac{1}{5} ...)

    The remainder term of this expansion is bounded by
    |R_n| = \frac{x^{2n+1}}{2n+1}

    The question is to work out how many terms you need to get pi accurate to 10 decimal places.
    So we set
    10^{-10} \geq \frac{4}{2n+1}
    (as 1^2n+1 = 1, and the expansion has been multiplied through by 4), to get n is approximately 2*10^{10}, which is how many terms you'd need to get pi accurate enough.
    However, the answer given was 5*10^{10}
    Can anyone see where we went wrong?
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    My guess it that to be accurate to 10 decimal places, you check to the 11th place, so change that 10^(-10) to 10^(-11) and then if you ignore the 4 you get the answer. I can't think why you ignore the 4 though.
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    Ooh, hadn't thought of that.
    Could you explain ignoring the 4 by saying that you need 5*10^10 terms to approximate pi/4; and then you can multiply what you get through by 4 as it's an exact constant and not part of the approximation?
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    That's what I thought, yeah. It's kind of the same as saying if you have 1/12 accurate to 10 places then you can multiply by 4 to get 1/3 to 10 places. Except when I do that on a calculator I only get 1/3 accurate to 9 places.
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    Okay, cheers
 
 
 
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Updated: November 15, 2008
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