Question

If \( a_1 = 1 \) and for \( n \geq 1 \), \[ a_{n+1} = \frac{1}{2} a_n + \frac{n^2 - 2n - 1}{n^2 (n+1)^2} \] then \[ \left| \sum_{n=1}^{\infty} \left( a_n - \frac{2}{n^2} \right) \right| \] is equal to

• A. 3
• B. 4
• C. 5
• D. 2

Hint:

For recurrence relations, carefully compute the first few terms and check for a pattern, then use the series summation technique to find the limit.

Answer 1

The Correct Option is D

Step 1: Understand the recurrence relation.
The recurrence relation is given by: \[ a_{n+1} = \frac{1}{2} a_n + \frac{n^2 - 2n - 1}{n^2 (n+1)^2} \] Using this recurrence relation, we calculate the values of \( a_n \) for successive values of \( n \). Step 2: Find the sum.
We need to compute the sum: \[ \sum_{n=1}^{\infty} \left( a_n - \frac{2}{n^2} \right) \] This sum converges to a value of 2.