Sum of Sequence Terms (Jee Mains)

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Sum of Sequence Terms

Let a₁ = 1and for n > 1, a_{n + 1}
= $\frac{1}{2}{a_n} + \frac{{{n^2} - 2n - 1}}{{{n^2}{{(n + 1)}^2}}}.$ Then $\left| {\sum\limits_{n = 1}^\infty {\left( {{a_n} - \frac{2}{{{n^2}}}} \right)} } \right|$ is equal to ______.

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