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Thread: Problem involving series

  1. #1
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    Problem involving series

    The following problem is in a section on convergence tests for series (comparison, ratio, root, integral, p-series, etc.) I've been stuck on it for about a day, so any help would be much appreciated.

    Suppose that $\displaystyle \Sigma a_{n}$ is a convergent series with positive terms. Let $\displaystyle \{S_{n}\}$ be the sequence of partial sums for $\displaystyle \Sigma a_{n}$. Let $\displaystyle S = lim \ S_{n}$ and let $\displaystyle \rho_{k} = \frac{a_{k+1}}{a_{k}}$. Suppose there is a number $\displaystyle N$ for which $\displaystyle \rho_{k} \ge \rho_{k+1}$ if $\displaystyle k \ge N$ and $\displaystyle \rho_{N} < 1$. Show that $\displaystyle S - S_{N} \le \frac{a_{N+1}}{1 - \rho_{N}}$.

    So far all I've been able to do is show that $\displaystyle S - S_{N} = a_{N+1}(1 + \rho_{N+1} + \rho_{N+1}\rho_{N+2} + \rho_{N+1}\rho_{N+2}\rho_{N+3} + ...)$ which means I need to show that $\displaystyle 1 + \rho_{N+1} + \rho_{N+1}\rho_{N+2} + \rho_{N+1}\rho_{N+2}\rho_{N+3} + ... \le \frac{1}{1 - \rho_{N}}$, but I'm not sure where to take it from here, or if this is even a good approach.
    Last edited by spoon737; May 9th 2012 at 01:37 PM.
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  2. #2
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    Re: Problem involving series

    So you know $\displaystyle 0 < \rho_N < 1$. What can you say about $\displaystyle \Sigma (\rho_N)^n$? Does it converge? Where to? You also know that $\displaystyle \rho_N \geq \rho_{N+1}$. So then what can you say about the sum that you found?
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  3. #3
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    Re: Problem involving series

    Ah, I see. Term for term, my series is less than or equal to a geometric series converging to $\displaystyle \frac{1}{1 - \rho_{N}}$. Thanks a billion!
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