Converging Series and Derivative Analysis for Positive Terms

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Homework Help Overview

The discussion revolves around the convergence of a series involving positive terms and the analysis of a related integral. Participants are exploring the connection between the series and its integral representation, as well as the implications of derivatives in this context.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants are examining the convergence of the series and its relationship to the integral. There are questions about the validity of the integral's form and its limits. Some participants are also questioning the assumptions made regarding the convergence to specific values.

Discussion Status

The discussion is active, with participants sharing different interpretations of the series and integral. Some have provided insights into the convergence of the series, while others are exploring the implications of the integral's evaluation. There is no explicit consensus yet, but various approaches are being considered.

Contextual Notes

There are references to specific values and forms from textbooks, which may influence the understanding of the convergence and integral evaluation. The discussion includes assumptions about the behavior of the functions involved as they approach certain limits.

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sum(1/(n(n^2-1)^(1/2)),n=2,infinity)
first derivative <0 for x>=2
I(1/(x(x^2-1)^(1/2)),x,2,infinity)
x=secT, dx=secTtanT
I(secTtanT/(secTtanT),T) ?
 
Last edited:
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What is your question?
 
my book is showing this series converging to pi/6
 
ok..this integral is in basic form of I(1/(u(u^2-a^2)^(1/2)),u)=1/a sec^-1(u/a)+C
a=1
sec^-1 u
lim sec^-1x as x-> infinity =pi/2 and sec^-1 2=pi/3
pi/2-pi/3 =pi/6
 

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