2025 NJC P1 Q10

A sequence of numbers \(u_1, u_2, u_3, \dots\) has a sum \(S_n\), where \(S_n = \sum_{r=1}^{n} u_r\). It is given that \(S_n = A - \frac{2}{(n+1)!}\), where \(A\) is a non-zero constant.

1. Find the value of \(A\) if \(u_1 = 1\).[1]
2. Show that \(u_n = \frac{2}{(n+1)[(n-1)!]}\) for \(n \ge 1\).[3]
3. Find a recurrence relation in the form \(u_{n+1} = [\mathrm{f}(n)]u_n\).[2]
4. Explain why \(S_n\) converges as \(n \to \infty\).[1]

5. Hence, find the least value of \(m\) such that the sum of the infinite series

   \(u_m + u_{m+1} + u_{m+2} + \dots\)

   does not exceed \(10^{-10}\).

   [3]