[tex] \sum_{j=k}^{\infty}\left\{{\frac{1}{b_{j}}-\frac{1}{b_{j+1}}}\right\}=\frac{1}{b_{k}}[/tex](adsbygoogle = window.adsbygoogle || []).push({});

This series holds for all real monotone sequences of [tex] b_j [/tex].

So if I were to carry this series out to say n I end up with a partial sum that looks like:

[tex] S_n=\frac{1}{b_k}-\frac{1}{b_{k+(n+1)}} [/tex]

Now as n goes to infinity we are left with just [tex] b_k [/tex]. This of course implies that [tex]\frac{1}{b_{k+(n+1)}}[/tex] goes to zero as n goes to infinity. So does this mean that the monotone sequence [tex]b_j[/tex] must equal {1,2,3,4,5,...,j} ? If not what exactly are the constraints on [tex]b_j[/tex] to make that series an identity?

Thanks for the help everyone.

JTB

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# Homework Help: Quick question about a telescoping series

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