Question 7 - A-Level Maths

Prove by induction that, for all integers $n \geqslant 1$, $$\frac { 3 } { 1 ^ { 2 } \times 2 ^ { 2 } } + \frac { 5 } { 2 ^ { 2 } \times 3 ^ { 2 } } + \frac { 7 } { 3 ^ { 2 } \times 4 ^ { 2 } } + \ldots + \frac { 2 n + 1 } { n ^ { 2 } ( n + 1 ) ^ { 2 } } = 1 - \frac { 1 } { ( n + 1 ) ^ { 2 } }$$
Find the smallest integer $n$ for which the sum of the series differs from 1 by less than $10 ^ { - 5 }$.