MathDB

For a positive integer

n

, consider the equation \frac{1}{x\minus{}1}\plus{}\frac{1}{4x\minus{}1}\plus{}\cdots\plus{}\frac{1}{k^2x\minus{}1}\plus{}\cdots\plus{}\frac{1}{n^2x\minus{}1}\equal{}\frac{1}{2}. (a) Prove that, for every

n

, this equation has a unique root greater than

1

, which is denoted by

x_n

. (b) Prove that the limit of sequence

(x_n)

4

n

approaches infinity.

Problem Statement