Series and Partial Sums
A series is a sequence of partial sums.
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Study Series and Partial Sums
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 \[\sum_{k=1}^\infty a_k:=\lim_{n\to\infty}s_n.\]
Remark 2. In the literature, the symbol \(\sum_{k=1}^\infty a_k\) is also called
series. So this symbol has a twofold meaning, namely the limit of the
series (if existent) and the series itself. At the accordant places of
this manuscript, the concrete meaning will be clear from the
context.
The above definition formally does not include infinite sums of kind
\[\sum_{k=0}^\infty
a_k,\qquad\sum_{k=2}^\infty a_k,\;\;\;\text{ or }\;\;\;
\sum_{k=n_0}^\infty a_k\;\text{ for some }n_0\in\mathbb{N}.\]
However, their meaning is straightforward to define and we will call
these expressions infinite series, too.
Courtesy of Marcus Waurick. Well-defined & Wonderful podcast, marcus-waurick.de.
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