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romano · Riemann Hypothesis Offline Joined: 10 Jun 2004 Posts: 304

n is a positive integer. Show that the equation 1/(x - 1) + 1/(22x - 1) + ... + 1/(n2x - 1) = 1/2 has a unique solution xn > 1. Show that as n tends to infinity, xn tends to 4.

**Posted:** Fri Feb 04, 2005 6:46 pm

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jmerry

Could you clarify your expression? What do you mean by "n2x"? Should that be $n^2x$ ?

**Posted:** Fri Feb 04, 2005 7:50 pm

romano · Riemann Hypothesis Offline Joined: 10 Jun 2004 Posts: 304

Yeah , I meant $\frac{1}{x-1} + \frac{1}{2^2x-1} + ... + \frac{1}{n^2x-1 } =\frac{1}{2}$

**Posted:** Sat Feb 05, 2005 8:49 pm

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Moubinool

$f_n(x)= \frac{1}{x-1} + \frac{1}{2^2x-1} + ... + \frac{1}{n^2x-1 }$

is defined on $]1,+\infty[$

$f'_n(x)<0$ so $f_n$ is strictly decreasing
$\lim_{1^{+}}f_n=+\infty ,\; \lim_{+\infty}f_n=0$
$f_n$ is continous there exist one $x_n>1$ such that $f_n(x_n)=1/2$

Let $g_n(x)=f_n(x)-1/2<g_{n+1}(x)$

$g_{n}(x_{n+1})<g_{n+1}(x_{n+1})=0=g_n(x_n)$

but $g_n$ is decreasing so $x_{n+1}>x_n$

not finish...

**Posted:** Sun Feb 13, 2005 1:11 pm