A series is a sequence of partial sums.

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Concept |
Content |

Sequences |
These object are needed to define limits later on. |

Convergence |
Convergent sequences have a well-defined limit. |

Sums and Products |
An important shorthand notation for calculations. |

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The topic of this part are “infinite sums” of the form \[\sum_{k=1}^\infty a_k\] for some sequence
\((a_n)_{n\in\mathbb{N}}\). Before we
present a mathematically precise definition, we present “a little
paradoxon” that aims to show that one really has to be careful with
series. Consider the case where \((a_n)_{n\in\mathbb{N}}=((-1)^n)_{n\in\mathbb{N}}\).
On the one hand we can compute \[\begin{aligned}
\sum_{k=1}^\infty
(-1)^k=&\,-1+1-1+1-1+1-1+1-1+1-\ldots\\[-0.4cm]=&\,(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+\ldots\\=&\,0+0+0+0+0+\ldots=0
\end{aligned}\] and on the other hand

\[\begin{aligned}
\sum_{k=1}^\infty
(-1)^k=&\,-1+1-1+1-1+1-1+1-1+1-\ldots\\[-0.4cm]=&\,-1+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+\ldots\\=&\,-1+0+0+0+0+0+\ldots=-1.
\end{aligned}\] This is a very dramatic contradiction! To exclude
such awkward phenomena, we have to use a precise mathematical definition
of “infinite sums”.

**Definition 1** (Infinite series). *Let \((a_n)_{n\in\mathbb{N}}\) be a sequence in
\(\mathbb{K}\). Then the sequence \((s_n)_{n\in\mathbb{N}}\) defined by \[s_n:=\sum_{k=1}^na_k\] is called
**infinite series* (or just “*series*”). The sequence
element \(s_n\) is called *\(n\)-th partial sum of \((a_n)_{n\in\mathbb{N}}\)*. The series
is called *convergent* if \((s_n)_{n\in\mathbb{N}}\) is convergent. In
this case, we write \[\sum_{k=1}^\infty
a_k:=\lim_{n\to\infty}s_n.\]

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