Sum of Series: 1/6 | Homework Statement and Solution

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Homework Help Overview

The problem involves finding the sum of a series defined by the expression \(\sum^{\infty}_{n=3}\frac{1}{(2n-3)(2n-1)}\). The subject area pertains to series and convergence in mathematical analysis.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to identify the nature of the series and expresses difficulty in applying known series techniques. Some participants suggest using partial fraction decomposition as a potential approach. There is a discussion about whether to split the series into two summations and how to proceed with the expansion of the series.

Discussion Status

The discussion is ongoing, with participants providing guidance on how to manipulate the series. There is a focus on understanding the structure of the series and the implications of expanding it. No explicit consensus has been reached, but there are indications of productive exploration of the problem.

Contextual Notes

Participants note the original poster's uncertainty about the series' characteristics and the provided answer from a textbook, which may influence their approach. There is an acknowledgment of the learning curve associated with recognizing series manipulation techniques.

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Homework Statement


Find the sum of the series:
[tex]\sum^{\infty}_{n=3}\frac{1}{(2n-3)(2n-1)}[/tex]

Homework Equations


N/A

The Attempt at a Solution


This isn't geometric, I can't get it from any common Maclaurin series (as far as I can work out). The book I have tells me the answer is 1/6, I'm sure I'm doing something stupid. Thanks in advance for any help!
 
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Try breaking up the summand using partial fraction decomposition, and then expand the series.
 
OK, so I have [tex]\sum^{\infty}_{n=3}\frac{1}{2(2n-3)}-\frac{1}{2(2n-1)}[/tex], but I'm not sure what you mean by "expanding the series"; should I split it into two summations? And if so, where do I go from there?
 
No, not two series - just one. Expanding the series means writing out the sum of terms, starting with the one for n = 3. Look at the sequence of partial sums, Sk, the sum of the terms from n = 3 to n = k.
 
*hits self in head* Wow! Thanks a ton, that should have been obvious.
 
These things are obvious only after you have done them a time or two.
 
Yah, well at least now when I see it again I'll know what to do! Thanks again for the help.
 

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