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# Series Cheat Sheet

## Definition

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:

\[s_n = \sum_{k = 0}^{n} a_k\]

### Convergence

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.

## Series of functions

Let $f_k(x)$ be a sequence of functions. A series of functions is defined as:

\[\sum_{k = 0}^{\infty} f_k(x)\]

### Convergence

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].

## References

- [1] Real Analysis (2nd edition) - Jay Cummings