Quotient criterion

The quotient criterion is a mathematical convergence criterion for series . It is based on the majorant criterion , that is, a complex series is rated upwards by a simple, here the geometric series . The geometric series converges precisely when the amount of the following terms decreases, i.e. the (constant) quotient of two successive terms is less than 1. If another series decreases at least as quickly from a certain element, i.e. if the quotient is smaller or equal , then this is also convergent. Divergence can also be demonstrated with the quotient criterion. If the quotient is always greater than or equal to 1, the amount of the following terms does not decrease. Since these then do not form a zero sequence , the series is divergent. ${\displaystyle q}$ ${\displaystyle q}$

The quotient criterion was developed by the mathematician and physicist Jean-Baptiste le Rond d'Alembert , in whose honor this mathematical statement is also called d'Alembert's convergence criterion .

statement

Decision tree for the quotient criterion

A series with real or complex summands is given for almost all . Is there a such that for almost all true ${\displaystyle S:=\sum _{n=0}^{\infty }a_{n}}$ ${\displaystyle a_{n}\neq 0}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle q<1}$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle \left|{\frac {a_{n+1}}{a_{n}}}\right|\leq q<1,}$

so the series is absolutely convergent . However, this applies to almost everyone ${\displaystyle n\in \mathbb {N} }$

${\displaystyle \left|{\frac {a_{n+1}}{a_{n}}}\right|\geq 1}$,

so the series is divergent.

It must not strive towards 1 from below. If, on the other hand, only applies, i.e. the quotient can come as close as desired to 1, the quotient criterion does not provide any information about the convergence or the divergence. ${\displaystyle \left|{\frac {a_{n+1}}{a_{n}}}\right|}$ ${\displaystyle \left|{\frac {a_{n+1}}{a_{n}}}\right|<1}$

In the case of convergence must of be independent. ${\displaystyle q}$ ${\displaystyle n}$

Examples

• We look at the series and check it for convergence. Using the quotient criterion, we get ${\displaystyle \sum _{n=0}^{\infty }{\frac {5+n}{10^{n}}}}$

${\displaystyle \left|{\frac {a_{n+1}}{a_{n}}}\right|={\frac {5+(n+1)}{10^{n+1}}}\cdot {\frac {10^{n}}{5+n}}={\frac {1}{10}}\cdot {\frac {6+n}{5+n}}\leq {\frac {3}{25}}<1}$.

Hence the series is convergent.

• We look at the series and check it for convergence. We obtain ${\displaystyle \sum _{n=1}^{\infty }{\frac {n!}{2^{n}}}}$

${\displaystyle \left|{\frac {a_{n+1}}{a_{n}}}\right|={\frac {(n+1)!}{2^{n+1}}}\cdot {\frac {2^{n}}{n!}}={\frac {n+1}{2}}\geq 1}$.

So this series is divergent.

• We want to determine the radius of convergence of the power series for complex numbers . For the series is obviously convergent to 0, so let us have ${\displaystyle \sum _{n=1}^{\infty }{\frac {n!}{n^{n}}}z^{n}}$ ${\displaystyle z\in \mathbb {C} }$ ${\displaystyle z=0}$ ${\displaystyle z\not =0}$

${\displaystyle \left|{\frac {a_{n+1}}{a_{n}}}\right|=\left|{\frac {(n+1)!n^{n}}{n!(n+1)^{n+1}}}{\frac {z^{n+1}}{z^{n}}}\right|=\left({\frac {1}{1+{\frac {1}{n}}}}\right)^{n}|z|\,\,{\overset {n\to \infty }{\longrightarrow }}\,\,{\frac {|z|}{e}}}$.

So the radius of convergence is Euler's number . ${\displaystyle e}$

• An example of the inapplicability of the quotient criterion is the general harmonic series . It applies ${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{\alpha }}}}$

${\displaystyle \left|{\frac {a_{n+1}}{a_{n}}}\right|=\left({\frac {n}{n+1}}\right)^{\alpha }<1}$.

For is the general harmonic series divergent, for convergent; however, the quotient criterion cannot distinguish the two cases. ${\displaystyle \alpha =1}$ ${\displaystyle \alpha >1}$

Proof idea

The case of convergence follows with the majorant criterion from the convergence of , a geometric series . The criterion for divergence follows from the fact that the terms can not form a zero sequence because of this. ${\displaystyle \sum _{n=0}^{\infty }q^{n}}$ ${\displaystyle 0<\left|a_{n}\right|\leq \left|a_{n+1}\right|}$

Special cases

Exist , thus providing the ratio test ${\displaystyle L:=\lim _{n\to \infty }\left|{\frac {a_{n+1}}{a_{n}}}\right|}$

1. for absolute convergence, ${\displaystyle L<1}$
2. for divergence, ${\displaystyle L>1}$
3. for no convergence statement. ${\displaystyle L=1}$

Using the Limes superior and Limes inferior , the quotient criterion can be formulated as follows:

1. If the series is absolutely convergent, ${\displaystyle \limsup _{n\to \infty }\left|{\frac {a_{n+1}}{a_{n}}}\right|<1}$
2. is , the series is divergent, ${\displaystyle \liminf _{n\to \infty }\left|{\frac {a_{n+1}}{a_{n}}}\right|>1}$
3. is , no statement of convergence can be made. ${\displaystyle \liminf _{n\to \infty }\left|{\frac {a_{n+1}}{a_{n}}}\right|\leq 1\leq \limsup _{n\to \infty }\left|{\frac {a_{n+1}}{a_{n}}}\right|}$

In contrast to the root criterion , the limes inferior rather than the limes superior must be used for the divergence criterion .

Modified quotient criterion

In addition to the "usual" quotient criterion, there are also the following versions (see also Raabe's criterion ): Be a sequence with genuinely positive terms. If now ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$

${\displaystyle \exists d>1,n_{0}\in \mathbb {N} :\forall n\geq n_{0}:{\frac {a_{n+1}}{a_{n}}}\leq 1-{\frac {d}{n}}}$,

so it holds that is convergent. ${\displaystyle \sum _{n=1}^{\infty }a_{n}}$

On the other hand is

${\displaystyle \exists n_{0}:\forall n\geq n_{0}:{\frac {a_{n+1}}{a_{n}}}\geq 1-{\frac {1}{n}}}$,

so follows:

${\displaystyle \sum _{n=1}^{\infty }a_{n}}$diverges against . ${\displaystyle \infty }$

Applications

With the quotient criterion, for example, the convergence of the Taylor series for the exponential function and for the sine and cosine functions can be shown.

literature

• Otto Forster : Analysis I differential and integral calculus of a variable. Rowohlt, Hamburg 1976.
• Konrad Knopp : Theory and Application of Infinite Series. 6th edition. Springer, 1996, ISBN 3-540-59111-7 ( online , 1964 edition).
• Peter Hartmann: Mathematics for Computer Scientists. 4th edition. Vieweg, 2006, ISBN 3-8348-0096-1 , p. 254.

Web links

Wikibooks: Math for non-freaks: Quotient criterion  - learning and teaching materials

• Mathematics Online Lexicon (Definition and Proof)

Individual evidence

1. ^ Wilhelm Merz: Mathematics for engineers and natural scientists . Springer Spectrum, Berlin 2013, ISBN 978-3-642-29979-7 , pp. 170 .
2. ↑ Harro Heuser : Textbook of Analysis . Part 1. 11th edition. BG Teubner, Stuttgart 1994, ISBN 3-519-42231-X , p. 205 f .