Absolutely convergent series

An absolute convergent series is a term used in analysis . It is a tightening of the concept of the convergent series . For the absolutely convergent series, some properties of finite sums remain valid, which are generally false for the larger set of the convergent series.

definition

A real-valued or complex-valued series is called absolutely convergent if the series of absolute values ${\displaystyle \textstyle \sum _{n=0}^{\infty }a_{n}}$

${\displaystyle \sum _{n=0}^{\infty }|a_{n}|<\infty }$

converges.

This definition is also generalized to normalized spaces : A series in a normalized space is called absolutely convergent if the series of norms converges.

Examples

• Convergent series, the summands of which are almost all non-negative, are absolutely convergent.
• The series

${\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n^{2}}}}$

is because

${\displaystyle \sum _{n=1}^{\infty }\left|{\frac {(-1)^{n-1}}{n^{2}}}\right|=\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}}$

absolutely convergent.

• The power series of the exponential function

${\displaystyle \exp(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}}$

is absolutely convergent for every complex . ${\displaystyle z}$

• In general, a real or complex power series is absolutely convergent inside its convergence circle.
• The alternating harmonic series

${\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n}}}$

is convergent to . But it is not absolutely convergent, because when you check the defining property you get ${\displaystyle \ln(2)}$

${\displaystyle \sum _{n=1}^{\infty }\left|{\frac {(-1)^{n-1}}{n}}\right|=\sum _{n=1}^{\infty }{\frac {1}{n}}}$,

thus the ordinary harmonic series. This is definitely divergent against . ${\displaystyle \infty }$

properties

Every absolutely convergent series is (unconditionally) convergent . This applies to both real-valued and complex-valued series. More generally: In finite-dimensional spaces, unconditionally convergent is synonymous with absolutely convergent.

But there are series that are convergent but not absolutely convergent; they are considered to be conditionally convergent . In infinite-dimensional spaces there are even necessarily convergent series that do not absolutely converge.

Some convergence criteria for series, such as the root criterion and the quotient criterion , require absolute convergence.

Rearrangements

→ Main article : Rearrangement of rows

An essential property of absolutely convergent series is that, as with finite sums, the summands can be interchanged at will: Every rearrangement of an absolutely convergent series , i.e. H. every row that results from rearrangement of the series members of is convergent and converges to the same limit value as . This is exactly the opposite of convergent, but not absolutely convergent series : There is always a rearrangement of that diverges. ${\displaystyle s}$ ${\displaystyle s}$ ${\displaystyle s}$ ${\displaystyle t}$ ${\displaystyle t}$

If the series is real-valued, the following, even sharper statement ( Riemann rearrangement theorem ) applies : For every given number there is a rearrangement of the series which converges to (improperly). The reason is easy to give, we limit ourselves to the case . The summands are arranged in two sequences ${\displaystyle t}$ ${\displaystyle S\in \mathbb {R} \cup \{\pm \infty \}}$ ${\displaystyle t}$ ${\displaystyle S}$ ${\displaystyle S\neq \pm \infty }$

${\displaystyle a_{1}\geq a_{2}\geq \dotsb \geq a_{n}\geq \dotsb >0>\dotsb \geq b_{n}\geq \dotsb \geq b_{2}\geq b_{1}}$

(summands that are equal to zero are omitted). Now one adds up sequence elements until it is exceeded, then (negative) sequence elements off until it falls below again, then off again, etc. The method is feasible because and diverge (otherwise the original series would be absolutely convergent), and the rearranged series Series converges against . ${\displaystyle (a_{n})}$ ${\displaystyle S}$ ${\displaystyle (b_{n})}$ ${\displaystyle S}$ ${\displaystyle (a_{n})}$ ${\displaystyle \sum a_{n}}$ ${\displaystyle \sum b_{n}}$ ${\displaystyle S}$

Generalizations

The concept of absolute convergence can be generalized to standardized spaces . A sequence of elements of a standardized space is given . The corresponding row is through ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (X,\|\cdot \|)}$ ${\displaystyle (s_{n})_{n\in \mathbb {N} }}$

${\displaystyle s_{n}:=\sum _{\nu =1}^{n}x_{\nu }}$

Are defined. The series is called absolutely convergent if it converges. ${\displaystyle \textstyle \sum _{\nu =1}^{\infty }\|x_{\nu }\|}$

If a Banach space is complete , then every absolutely convergent series is also convergent. In fact, the reverse is also true: If a normalized vector space and every absolutely convergent series convergent, then it is complete, i.e. a Banach space. ${\displaystyle X}$ ${\displaystyle (X,\|\cdot \|)}$ ${\displaystyle X}$

In any complete metric space there is a related result. A sequence is at least convergent if the sum ${\displaystyle \left(s_{n}\right)_{n\in \mathbb {N} }}$

${\displaystyle \sum _{\nu =1}^{\infty }d\left(s_{\nu -1},s_{\nu }\right)}$

converges. Since yes in the above example , the absolute convergence results as a special case. ${\displaystyle d\left(s_{\nu -1},s_{\nu }\right)=\|x_{\nu }\|}$

literature

• Avner Friedman: Foundations of Modern Analysis. Dover, New York 1970. ISBN 0-486-64062-0 .
• Konrad Knopp: Theory and Application of Infinite Series . 5th edition, Springer Verlag 1964, ISBN 3-540-03138-3 .

Web links

Wikibooks: Math for Non-Freaks: Absolute Convergence of a Series  - Learning and Teaching Materials