Solve Series Proof Homework: 0<b<1 Convergence & Limit

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SUMMARY

The discussion focuses on proving the convergence of the series \(\sum^{n}_{r=1} \left(\frac{1}{rb} - \frac{n(1-b)}{1-b}\right)\) as \(n\) approaches infinity, where \(0 < b < 1\). The limit is denoted as \(\beta = \beta(b)\). Additionally, it addresses the equation \(\sum^{\infty}_{n=1} \frac{(-1)^{n-1}}{nb} + \beta(2(1-b) - 1) = 0\). Participants suggest using the integral test as a method for solving the problem.

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Homework Statement



Let 0<b<1, show that [tex]\sum[/tex][tex]^{n}_{r=1}[/tex] (1/rb - [tex]\frac{n<sup>1-b</sup>}{1-b}[/tex]) converges as n goes to infinity and denote the limit by [tex]\beta[/tex] = [tex]\beta[/tex](b).

Also, show that [tex]\sum[/tex][tex]^{infinity}_{n=1}[/tex] [tex]\frac{(-1)<sup>n-1</sup>}{n<sup>b</sup>}[/tex] + [tex]\beta[/tex](21-b - 1) = 0

Homework Equations


The Attempt at a Solution



Absolutely clueless!

**Sorry for the bad formatting, for I am new to PF.
 
Last edited:
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Try an integral test...
 

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