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Math Help - Alternating Series Theorem (remainder formula)

  1. #1
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    Alternating Series Theorem (remainder formula)

    The Alternating Series Estimation theorem states:

    If s= [summation](-1)^n*b_n from n=1 to infin. is the sum of a convergent, alternating series that satisfies:
    a) 0 < b_n+1 < b_n
    b) lim b_n -> 0 as n-> infin

    then:

    |R_n|= s - s_n <= b_n+1


    So the problem I'm given is this:

    How many terms of the series do I need to add in order to find the sum to the indicated accuracy?

    [summation](-2)^n/n! ; error < 0.01

    Okay, so I can use the theorem mentioned above because this series is both alternating and convergent (the positive part of the sum is decreasing for all n, and the lim 1/n! -> 0 as n-> infin, satisfying both conditions for convergence).

    So... |R_n| <= b_n+1 < 1/100... right?

    b_n+1 = 1/(n+1)! < 1/100

    So how do I solve for n? Looking at the solution for this problem, it seems like the person kept guessing random values of n until (-2)^n/n! was less than 1/100, which I don't really understand. The answer ended up being 7. That is, the sum of 7 terms to keep within the given accuracy.

    I guess I don't know how to do these kinds of problems when the "negative part of the series" is something other than (-1)^n.
    Last edited by Hikari; July 22nd 2009 at 01:56 PM.
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  2. #2
    MHF Contributor Calculus26's Avatar
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    first of all bn = 2^n/n! since (-2)^n = (-1)^n 2^n

    you want 2^(n+1)/[(n+1)!] < .01

    There is no analytic way of solving this--simply make a table of values

    using your calculator and look for the smallest n

    for which 2^(n+1)/[(n+1)!] < .01

    Or if you prefer crank it aout by hand try n= 2 if that doesn't work try n = 3 etc.
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  3. #3
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    Eureka! Thanks a bunch!
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  4. #4
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    Just re-writing with latex:

    If s = \sum_{n=0}^{\infty} (-1)^n b_n is the sum of a convergent, alternating series that satisfies:

    a) 0 < b_n+1 < b_n

    b) \lim_{n=\infty} b_n = 0

    then:

    |R_n|= s - s_n <= b_n+1

    Problem: How many terms of the series do I need to add in order to find the sum to the indicated accuracy?

    \sum_{n=0}^{\infty} \frac{(-2)^n}{n!}; error <0.01
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