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Math Help - sequence, show convergence.

  1. #1
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    sequence, show convergence.

    here it is:ImageShack - Image Hosting :: surehu5.jpg

    for n >= 1 is decreasing, and positive for all n. Show {An} is convergent.

    need help please.
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  2. #2
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    Could you please write the formulas in the forum using LaTeX in the future as probably very few people will check your handwriting on ImageShack...

    The series: a_1=1, a_{n+1}=a_n\cdot\left(1-\frac {a_n}{2n}\right).

    So we have to show that 0<a_n<1 and it is decreasing.
    Using induction we prove that it's decreasing and positive. We know that 1=a_1>a_2=\frac 12. Suppose that a_{n-1}\cdot\left(1-\frac {a_{n-1}}{2n-2}\right)=a_n<a_{n-1} and a_{n-1}>0, a_n>0. We prove that: a_n\cdot\left(1-\frac {a_n}{2n}\right)=a_{n+1}<a_n which is equivalent to: 0<\frac {a_n}{2n} by using the assumption a_n>0 and which is true. We only need to prove that a_{n+1}>0 or equivalently \frac {a_n}{2n}<1\Leftrightarrow a_n<2n. We know by assumption that a_{n-1}<2n-2 and because a_n<a_{n-1} we're done \Box

    So we know that a_n is bounded and monotone therefore convergent.
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