absolutely convergent

Absolutely convergent series / Absolutely summable sequence

Let $(a_n{)}_{n\in \mathbb{N}}\subset \mathbb{C}$ be a sequence such that

$$\sum _{k=1}^{\infty }|a_k|<\infty :⟺\exists C\in \mathbb{R}\forall n\in \mathbb{N}:\sum _{k=1}^n|a_k|\le C,$$

i.e. we say that the series ${\sum }_{k=1}^{\infty }$ is absolutely convergent, or the sequence $(a_n{)}_{n\in \mathbb{N}}$ is absolutely summable, if the series of absolute values of $a_n$ is bounded.

For non-negative sequences, absolute summability and summability are the same.

Absolutely convergent series are also convergent.

Proof: Let ${\sum }_{k=1}^{\infty }{a}_k$ be a absolutely convergent series and $ϵ>0$. By the cauchy criterion, there exists some $n_0\in N$ such that for all $m>n\ge n_0$ we have ${\sum }_{k=n}^m|a_k|<ϵ$ (the bars are on the inside because we defined the series to be absolutely convergent).
Then the triangle inequality yields

$$|\sum _{k=n}^m{a}_k|\le \sum _{k=n}^m|a_k|<ϵ$$

which, again by the cauchy criterion ($|{\sum }_{k=n}^m{a}_k|<ϵ$), implies that ${\sum }_{k=1}^{\infty }{a}_k$ is convergent.