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Thread: sequence, show convergence.

  1. #1
    Senior Member
    Jan 2007

    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
    Senior Member
    Nov 2007
    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: $\displaystyle a_1=1$, $\displaystyle a_{n+1}=a_n\cdot\left(1-\frac {a_n}{2n}\right)$.

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

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