The sum of the first $n$ terms of a sequence, $u_r$, is given by \[\sum_{r=1}^{n} u_r = 1 - \fra

Question

The sum of the first $n$ terms of a sequence, $u_r$, is given by $$\sum_{r=1}^{n} u_r = 1 - \frac{n}{(n+1)!}.$$ Find $u_n$ in terms of $n$, expressing your answer as a single algebraic fraction.[2] Explain why $\displaystyle \sum_{r=5}^{n} u_r < \frac{1}{30}$, for all $n \ge 5$.[2] Explain why $\displaystyle \sum_{r=1}^{\infty} u_r$ is a convergent series.[1]

Accepted Answer

(a) $u_n=\frac{n^2-n-1}{(n+1)!}$ (b) $\sum_{r=5}^{n}u_r=\frac{1}{30}-\frac{n}{(n+1)!}<\frac{1}{30}$ (c) $\sum_{r=1}^{\infty}u_r=1$

Sequence and Series - Problems involving inequalities

Sequence and Series - Problems involving inequalities

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