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Introduction to Series

Let
$\displaystyle\displaystyle\sum_{i=1}^\infty a_i$ be a
series, and let $\displaystyle s_n = \sum_{i=1}^n a_i$ be
its $n^{th}$ partial sum. We define $\displaystyle\sum_{i=1}^\infty a_i = \lim_{n \to \infty} s_n.$ If this limit exists and is finite and
equal to $s$, we say the series is convergent
and that $\displaystyle\sum_{i=1}^\infty a_i=s$.
Otherwise, we say the series is divergent.
