IAL 2022 June Q1 | 國際數學站

Given that

\begin{align*} \frac{2n + 1}{n^2 (n + 1)^2} \equiv\,& \frac{A}{n^2} + \frac{B}{(n + 1)^2}\\[2mm] \end{align*}

(a) determine the value of $A$ and the value of $B$

(1)

(b) Hence show that, for $n \geqslant 5$

\begin{align*} \sum_{r=5}^n \frac{2r + 1}{r^2 (r + 1)^2} =\,& \frac{n^2 + an + b}{c(n + 1)^2}\\[2mm] \end{align*}

where $a$ , $b$ and $c$ are integers to be determined.

(4)