- #1
Townsend
- 232
- 0
[tex] \sum_{j=k}^{\infty}\left\{{\frac{1}{b_{j}}-\frac{1}{b_{j+1}}}\right\}=\frac{1}{b_{k}}[/tex]
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
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