Is the Infinite Series of Pi/2 a Mathematical Marvel?

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Discussion Overview

The discussion centers around an infinite series purportedly equal to Pi/2, with participants exploring its formulation, potential proofs, and methods of evaluation. The conversation includes mathematical reasoning and attempts to clarify the series' structure.

Discussion Character

  • Mathematical reasoning
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents an infinite series that they believe sums to Pi/2, asking for proof or references.
  • Another participant suggests that the series may actually sum to 2, questioning the formulation and indicating a possible mistake in the original expression.
  • Clarification is sought regarding the sign of the (1/4) term, with a later participant asserting that it should be positive to yield Pi/2.
  • A corrected version of the series is provided, which involves the terms of the form ((2n-1)^2) - (1/4).
  • Partial fractions are recommended as a method to evaluate the series, with a specific decomposition provided by one participant.
  • Another participant expresses interest in exploring alternative methods to derive the series.

Areas of Agreement / Disagreement

Participants exhibit disagreement regarding the correct formulation of the series and its sum. While some suggest it sums to Pi/2, others propose it sums to 2, indicating that the discussion remains unresolved.

Contextual Notes

There are unresolved questions about the correct formulation of the series and the assumptions underlying the mathematical manipulations. The discussion also reflects differing interpretations of the series' convergence and evaluation methods.

the dude man
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Hey guys!

My friend showed me a infinite series on maple

infinity
---
\ ( ((n-1)^2) - (1/4) )^(-1) = (Pi/2)
/
---
n=1

Is anyone familar with this?
If so can you refer me to its proof? Or maybe post it.

Thanks
 
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Hrm. Unless I made a mistake, that sums to 2. You meant

<br /> \sum_{n = 1}^{+\infty} \frac{1}{ (n-1)^2 - \frac{1}{4}}<br />

right? Anyways, the usual trick to these is to use partial fractions.
 
Sorry i think the (1/4) is positive

that should give (Pi/2)

yes/no?
 
Sorry here's the correct series

infinity
----
\ (((2*n-1)^2) - (1/4))^(-1)
/
----
n=1
 
Use partial fractions, as hinted already.

\frac{1}{\left(2n-1\right)^{2}-\frac{1}{4}}=2\left(\frac{1}{4n-3}-\frac{1}{4n-1}\right)

Daniel.
 
That works but are there any other options?
 
Last edited:
Im interested in the ways you can obtain the series.
 

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