Mean Sequences

(2.51) if $b_{n} = \frac{1}{n} \sum_{i = 1}^{n} a_{i}$ then $(b_{n})$ is the arithmetic mean sequence of $(a_{n})$ and $n \to \infty lim (a_{n}) = n \to \infty lim (b_{n})$
(2.52) if $c_{n} = n \prod_{i = 1}^{n} a_{i}$ then $(c_{n})$ is the geometric mean sequence of a positive sequence $(a_{n})$ and $n \to \infty lim (a_{n}) = n \to \infty lim (c_{n})$
(q2.62) Stolz–Cesàro theorem - if $(b_{n})$ is a strictly monotone and divergent sequence, then $n \to \infty lim \frac{a _{n + 1} - a _{n}}{b _{n + 1} - b _{n}} = n \to \infty lim \frac{a _{n}}{b _{n}}$

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