Question 136
Consider the equation
$$
\frac{1}{x-1} + \frac{1}{4x-1} + \dots + \frac{1}{k^2x-1} + \dots + \frac{1}{n^2x-1} = \frac{1}{2}
$$
where $n$ is a positive integer parameter.
1/ Prove that for every positive integer $n$, the considered equation has a unique root greater than $1$, which is denoted by $x_n$.
2/ Prove that the sequence $\{x_n\}$ has limit $4$ (when $n$ tends to $+\infty$).
$$
\frac{1}{x-1} + \frac{1}{4x-1} + \dots + \frac{1}{k^2x-1} + \dots + \frac{1}{n^2x-1} = \frac{1}{2}
$$
where $n$ is a positive integer parameter.
1/ Prove that for every positive integer $n$, the considered equation has a unique root greater than $1$, which is denoted by $x_n$.
2/ Prove that the sequence $\{x_n\}$ has limit $4$ (when $n$ tends to $+\infty$).