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Let $(a_k)$ be a sequence. A series is the (infinite) sum of these values:
\[\sum_{k = 0}^{\infty} a_k\]The partial sum is defined as:
\[s_n = \sum_{k = 0}^{n} a_k\]A series converges to a finite $L \in \mathbb{R}$ if the sequence of partial sums, $(s_n)$, converges.
A series converges absolutely if the series of its absolute summands, i.e.
\[\sum_{k = 0}^{\infty} \abs{a_k}\]converges.
Let $f_k(x)$ be a sequence of functions. A series of functions is defined as:
\[\sum_{k = 0}^{\infty} f_k(x)\]Define the partial sum:
\[s_n = \sum_{k = 0}^{n} f_k(x)\]A series of functions converges uniformily (or pointwise) if the sequence $(s_n)$ converges uniformily (or pointwise). Note that uniform and pointwise convergence only makes sense for series/sequences of functions.
A series of functions converges absolutely if the series
\[\sum_{k = 0}^{\infty} \abs{f_k(x)}\]converges, either uniformily or pointwise.
Weierstrass M-test. If $\abs{f_k(x)} \le M_k$ for all $k$ and $x \in A$ and the series $\sum_{n = 0}^\infty M_n$ converges, then
\[\sum_{k = 0}^{n} f_k(x)\]converges absolutely and uniformly [1].