Serlo: EN: Bounded series and convergence

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Bounded series with positive summands converge

As we learned in the chapter „Monotoniekriterium für Folgen“, monotonicity and boundedness combined are a sufficient criterion to show convergence of a sequence. This criterion is also applicable to series, since the meaning of a series is defined by the sequence of its partial sums. In the following, we will derive a convergence theorem for series that is based on the monotonicity and boundedness criterion for sequences.

We proceed in two steps by separately examining the monotonicity and boundedness of partial sums $s_n\equiv \sum _{k=1}^n{a}_k$ of some series $\sum _{k=1}^{\mathrm{\infty }}{a}_k$.

First we ask: Under what circumstances is the sequence $(s_n{)}_{n\in \mathbb{N}}$ of partial sums monotonically increasing? Starting with an arbitrary partial sum, we obtain the next one by adding the corresponding term of the series. Therefore the next partial sum is greater (or equal), whenever this summand is positive (or zero). If we require monotonicity for the partial sums, this means all terms of the series must be non-negative (except for the first one, which does not play the role of a difference between partial sums).

We now want to show this statement formally. The relation between two subsequent partial sums is Vorlage:Einrücken The sequence $(s_n{)}_{n\in \mathbb{N}}$ is monotonically increasing, if $s_{n+1}\ge s_n$ for all $n\in \mathbb{N}$. Using the relation we just derived, we find: Vorlage:Einrücken Hence the sequence of partial sums is monotonically increasing, if and only if $a_{n+1}\ge 0$ for all $n\in \mathbb{N}$. This simply means $a_n\ge 0$ for all $n\ge 2$.

Next, we address boundedness of the sequence $(s_n{)}_{n\in \mathbb{N}}$. A real sequence is bounded, if there exists some constant $b\in ℝ$, such that $s_n\le b$ for all $n\in \mathbb{N}$. With the $s_n$ being partial sums, we can picture this in the following way: Each partial sum $s_n$ is a truncation of the infinite sum $\sum _{k=1}^{\mathrm{\infty }}{a}_k$, where we drop all terms after $a_n$. Each partial sum is just a finite sum of finite terms, so it is clearly finite itself. Therefore it is not hard to find a common upper bound for some of the partial sums. However, to show boundedness of the sequence, we need to find an upper bound for all partial sums at once. So we are looking for a number that is greater than any of the truncated sums, no matter how late we do the truncation.

We summarize these observations in the following theorem:

Analogously, we can also proof a theorem for monotonically decreasing partial sums:

Application: Convergence of the hyperharmonic series Vorlage:Anker

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