A sequence of numbers $u_1, u_2, u_3, \dots$ has a sum $S_n$, where \(S_n = \sum_{r=1}^{n}

Question

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

Accepted Answer

(i) $A=2$ (ii) $u_n=\dfrac2{(n+1)(n-1)!}$ for $n\ge1$ (iii) $u_{n+1}=\dfrac{n+1}{n(n+2)}u_n$ (iv) $S_n\to2$ (v) $m=14$

2025 NJC P1 Q10

2025 NJC P1 Q10

Solution:

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