Alternating series

The title of this article is ambiguous. For Leonhard Euler's alternating series, see Alternating Series (Euler)

Alternating series are infinite series and as such belong in the mathematical branch of analysis .

definition

An alternating series ( English English alternating series ) is an infinite series, the series terms of real numbers exist, the alternate sign have.

So it's a series that is in the form

${\displaystyle \sum _{k=0}^{\infty }(-1)^{k}a_{k}}$   or   ${\displaystyle \sum _{k=1}^{\infty }(-1)^{k}a_{k}}$

can be represented, where the are. Often, in addition, requested that the result or monotonically decreasing should be. ${\displaystyle a_{k}\geq 0}$ ${\displaystyle (a_{k})_{k\in \mathbb {N} _{0}}}$ ${\displaystyle (a_{k})_{k\in \mathbb {N} }}$

Representation of constants using alternating series

Many constants in analysis have meaningful series representations and gain their interest not least from representations using alternating series. Here are some great examples - such as:

To the natural logarithm of 2

This is where one of the standard examples of alternating series that is repeatedly mentioned occurs, namely the alternating harmonic series

${\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k}}=1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+-\ldots =\ln 2}$,

which, in contrast to the ( divergent !) harmonic series, converges according to the Leibniz criterion .

To the Euler number

Another common example is the alternating series for the reciprocal of Euler's number . You have:

${\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k!}}=1-{\frac {1}{1!}}+{\frac {1}{2!}}-{\frac {1}{3!}}+{\frac {1}{4!}}+-\ldots ={\frac {1}{e}}}$.

To the circle number

Another standard example is the Leibniz series , which contains a series expansion of the circle number :

${\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}+-\ldots ={\frac {\pi }{4}}}$.

There are a number of other alternating series for the number of circles, such as

${\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k^{2}}}=1-{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}-{\frac {1}{4^{2}}}+{\frac {1}{5^{2}}}+-\ldots ={\frac {{\pi }^{2}}{12}}}$

and

${\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k^{4}}}=1-{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}-{\frac {1}{4^{4}}}+{\frac {1}{5^{4}}}+-\ldots ={\frac {{7\cdot \pi }^{4}}{720}}}$

and

${\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{(2k-1)^{3}}}=1-{\frac {1}{3^{3}}}+{\frac {1}{5^{3}}}-{\frac {1}{7^{3}}}+{\frac {1}{9^{3}}}+-\ldots ={\frac {{\pi }^{3}}{32}}}$.

To the root of 2

There are two examples for the root of the natural number that result from the binomial series , namely: ${\displaystyle 2}$ ${\displaystyle \sum _{k=1}^{\infty }(-1)^{k-1}{\frac {\binom {2k}{k}}{2^{2k}(2k-1)}}={\frac {1}{2}}-{\frac {1}{2\cdot 4}}+{\frac {1\cdot 3}{2\cdot 4\cdot 6}}+-\ldots ={\sqrt {2}}-1}$

and

${\displaystyle \sum _{k=0}^{\infty }(-1)^{k}{\frac {\binom {2k}{k}}{2^{2k}}}=1-{\frac {1}{2}}+{\frac {1\cdot 3}{2\cdot 4}}-{\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}+-\ldots ={\frac {1}{\sqrt {2}}}}$.

To the golden ratio

The golden number gives the following example: ${\displaystyle \Phi }$

${\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{k^{2}{\binom {2k}{k}}}}={\frac {1}{1\cdot 2}}-{\frac {1}{4\cdot 6}}+{\frac {1}{9\cdot 20}}-{\frac {1}{16\cdot 70}}+-\ldots =2\cdot {\ln(\Phi )}^{2}}$

The equation also proves the close connection with the Fibonacci numbers ${\displaystyle \left(f_{n}\right)_{n\in \mathbb {N} _{0}}=(1,1,2,3,5,8,13,21,34,55,89,144,233,\ldots )}$

${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{f_{k}}}=3{,}3598856662\ldots ={\sqrt {5}}\cdot \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{{\Phi }^{2n+1}-(-1)^{n}}}}$.

To the apéry constant

The Apéry constant , i.e. the function value of the Riemann Zeta function for the argument , also provides examples: ${\displaystyle x=3}$

${\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{k^{3}{\binom {2k}{k}}}}={\frac {1}{1\cdot 2}}-{\frac {1}{8\cdot 6}}+{\frac {1}{27\cdot 20}}-{\frac {1}{64\cdot 70}}+-\ldots ={\frac {2}{5}}\cdot \zeta (3)}$

The following series representation also applies:

${\displaystyle \sum _{k=1}^{\infty }\left((-1)^{k-1}\cdot {\sum _{m=1}^{\infty }{\frac {1}{km(k+m)}}}\right)=\sum _{k,m=1}^{\infty }{\frac {(-1)^{k-1}}{km(k+m)}}={\frac {5}{8}}\cdot \zeta (3)}$.

To the Catalan constant

The Catalan constant is even defined as an alternating series, namely as the following:

${\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k+1)^{2}}}=1-{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}-{\frac {1}{7^{2}}}+-\ldots =G}$

About the Cahen constant

Another example is Cahen's constant

${\displaystyle c=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{s_{k}-1}}=1-{\frac {1}{2}}+{\frac {1}{6}}-{\frac {1}{42}}+{\frac {1}{1806}}-{\frac {1}{3263442}}+-\ldots =0{,}6434105462\ldots }$

to be mentioned, whereby the sequence is defined by recursion : ${\displaystyle \left(s_{k}\right)_{k\in \mathbb {N} _{0}}}$

${\displaystyle s_{0}=2}$

${\displaystyle s_{k+1}={s_{k}}^{2}-s_{k}+1\;(k\in \mathbb {N} _{0})}$.

Closely related to Cahen's constant is the constant given by an alternating series

${\displaystyle c^{'}=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{s_{k}}}={\frac {1}{2}}-{\frac {1}{3}}+{\frac {1}{7}}-{\frac {1}{43}}+{\frac {1}{1807}}-{\frac {1}{3263443}}+-\ldots =2\cdot c-1=0{,}28682109258\ldots }$.

To the Euler-Mascheroni constant

A particularly remarkable example is provided by the Euler-Mascheroni constant represented by an alternating series using the function values ​​of the Riemann zeta function: ${\displaystyle \gamma }$

${\displaystyle \gamma =\sum _{k=2}^{\infty }(-1)^{k}{\frac {\zeta (k)}{k}}}$.

Other representations are also known, such as that of Vacca's formula :

${\displaystyle \gamma =\sum _{k=2}^{\infty }{\frac {(-1)^{k}}{k}}\cdot \lfloor {\frac {\ln k}{\ln 2}}\rfloor ={\frac {1}{2}}-{\frac {1}{3}}+{\frac {1}{2}}-{\frac {2}{5}}+{\frac {1}{3}}-{\frac {2}{7}}+{\frac {3}{8}}+-\ldots \ }$.

To a prime number constant

If one forms the corresponding alternating series from the reciprocal values ​​of the prime numbers , one obtains: ${\displaystyle \left(p_{k}\right)_{k\in \mathbb {N} }=\left(2,3,5,7,11,13,17,19,23,29,\ldots \right)}$

${\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k}}{p_{k}}}=-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{5}}+{\frac {1}{7}}-+\ldots =-0{,}2696063519\ldots }$.

For two constants treated by Ramanujan

The Indian mathematician Srinivasa Ramanujan found two alternating series to represent two constants in connection with the gamma function and the circle number , namely ${\displaystyle \Gamma }$ ${\displaystyle \pi }$

${\displaystyle \sum _{k=0}^{\infty }(-1)^{k}{\frac {{\binom {2k}{k}}^{2}}{2^{4k}}}={\frac {{\Gamma \left({\frac {1}{4}}\right)}^{2}}{\left(2\pi \right)^{\frac {3}{2}}}}}$

and

${\displaystyle \sum _{k=0}^{\infty }(-1)^{k}{\frac {{\binom {2k}{k}}^{3}}{2^{6k}}}=\left({\frac {\Gamma \left({\frac {9}{8}}\right)}{\Gamma \left({\frac {5}{4}}\right)\cdot \Gamma \left({\frac {7}{8}}\right)}}\right)^{2}}$.

To the integral of x to the power of x

The integral

${\displaystyle \int _{0}^{1}{x^{x}}\mathrm {d} x=\lim _{t\searrow 0}\int _{t}^{1}{x^{x}}\mathrm {d} x=0{,}7834305107\ldots }$

owns the representation

${\displaystyle \int _{0}^{1}{x^{x}}\mathrm {d} x=\sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{k^{k}}}=1-{\frac {1}{4}}+{\frac {1}{27}}-{\frac {1}{256}}+{\frac {1}{3125}}+-\ldots }$.

Representations of functions using alternating rows

Like the constants occurring in analysis, many real functions also have series representations using alternating series. There are a number of significant examples of this - such as:

To the logarithm function

The above example to the logarithm of can be generalized. Here 'dependent' for real numbers with the series expansion ${\displaystyle 2}$ ${\displaystyle x}$ ${\displaystyle -1<x\leq 1}$

${\displaystyle \ln(1+x)=\sum _{k=1}^{\infty }(-1)^{k-1}{\frac {x^{k}}{k}}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}+-\ldots }$,

from which for nonnegative (apparently) alternating series emerge. ${\displaystyle x}$

To the reciprocal function

An interesting example is provided by the geometrical series formed for real with ${\displaystyle x}$ ${\displaystyle |x|<1}$

${\displaystyle \sum _{k=0}^{\infty }(-1)^{k}x^{k}=\sum _{k=0}^{\infty }(-x)^{k}=1-x+x^{2}-x^{3}+-\ldots ={\frac {1}{1+x}}}$.

This forms an alternating series in the event that it is also absolutely convergent . The situation here is that the series sum is simply determined as the sum of the partial series formed only from the positive and only from the negative terms, i.e. as the difference between two series of all positive terms. ${\displaystyle x\geq 0}$

To the arctangent function

The above example for the Leibniz series can be generalized by means of the (alternating!) Arctangent series for real numbers with . The following applies here: ${\displaystyle x}$ ${\displaystyle -1\leq x\leq 1}$

${\displaystyle \arctan x=\sum _{k=0}^{\infty }(-1)^{k}{\frac {x^{2k+1}}{2k+1}}=x-{\frac {1}{3}}x^{3}+{\frac {1}{5}}x^{5}-{\frac {1}{7}}x^{7}+\ldots }$.

To sine and cosine

The Taylor series for the real sine and cosine functions are also among the important alternating series :

${\displaystyle \sin(x)=\sum _{k=0}^{\infty }(-1)^{k}{\frac {x^{2k+1}}{(2k+1)!}}=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+-\ldots \;(x\in \mathbb {R} )}$

${\displaystyle \cos(x)=\sum _{k=0}^{\infty }(-1)^{k}{\frac {x^{2k}}{(2k)!}}=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+-\ldots \;(x\in \mathbb {R} )}$

On the Riemann zeta function and the Dirichlet eta function

In connection with the above-mentioned alternating harmonic series, the following alternating series belongs as a further example, which is closely connected with the (already mentioned) Riemannian zeta function and which can be considered one of many examples of a Dirichlet series . Here you get, as GM Fichtenholz explains in his differential and integral calculus II , the representation for real numbers : ${\displaystyle x\in \mathbb {R} ^{>1}}$

${\displaystyle \eta (x)=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k^{x}}}=1-{\frac {1}{2^{x}}}+{\frac {1}{3^{x}}}-{\frac {1}{4^{x}}}+-\ldots =\left(\ 1-{\frac {1}{2^{x-1}}}\right)\cdot \zeta (x)}$.

In a similar way, one has the representation for real numbers with ${\displaystyle x}$ ${\displaystyle 0<x<1}$

${\displaystyle \zeta (x)={\frac {-1}{2^{1-x}-1}}\cdot \eta (x)}$

and then even

${\displaystyle \zeta (x)=\lim _{n\to \infty }\left(\sum _{k=1}^{n}{\frac {1}{k^{x}}}-{\frac {n^{1-x}}{1-x}}\right)}$.

To the Dirichlet beta function

The Catalan constant mentioned above also belongs to a functional example. It is the Dirichlet beta function , which is an alternating series for real numbers ${\displaystyle G}$ ${\displaystyle x\in \mathbb {R} ^{>0}}$

${\displaystyle \beta (x)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k+1)^{x}}}=1-{\frac {1}{3^{x}}}+{\frac {1}{5^{x}}}-{\frac {1}{7^{x}}}+{\frac {1}{9^{x}}}+-\ldots }$

can be represented.

To the Bessel functions

In connection with the Bessel differential equation , the Bessel functions -th order 1st genus ${\displaystyle n}$ occur, which for real numbers always alternate series of the form ${\displaystyle J_{n}}$ ${\displaystyle x\in \mathbb {R} }$

${\displaystyle J_{n}(x)=\sum _{k=0}^{\infty }{(-1)^{k}\cdot {\frac {\left({\frac {x}{2}}\right)^{n+2k}}{k!\cdot \Gamma (n+k+1)}}}={\frac {x^{n}}{2^{n}\cdot \Gamma (n+1)}}\cdot \left(1-{\frac {x^{2}}{2\cdot (2n+2)}}+{\frac {x^{4}}{2\cdot 4\cdot (2n+2)\cdot (2n+4)}}-+\ldots \right)}$

deliver.

Example of a divergent alternating series

An example of a divergent alternating series is

${\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k+1}\cdot (k+1)}{k}}=2-{\frac {3}{2}}+{\frac {4}{3}}-{\frac {5}{4}}+-\ldots }$,

It should be noted that the sequence is monotonically falling, but has the limit value . ${\displaystyle \left({\frac {k+1}{k}}\right)_{k=1,2,3,\dots ,\infty }}$ ${\displaystyle 1}$

literature

• Martin Barner , Friedrich Flohr : Analysis I (=  de Gruyter textbook ). 5th revised edition. Walter de Gruyter & Co. , Berlin, New York 2000, ISBN 3-11-016778-6 .
• IN Bronstein , KA Semendjajew , Gerhard Musiol, Heiner Mühlig (Hrsg.): Taschenbuch der Mathematik . 10th, revised edition. Europa-Lehrmittel, Haan-Gruiten 2016, ISBN 978-3-8085-5790-7 .
• Claudio Canuto, Anita Tabacco: Mathematical Analysis I (=  UNITEXT - La Matematica per il 3 + 2 . Band 84 ). 2nd Edition. Springer International Publishing Switzerland , Cham, Heidelberg, New York, Dordrecht, London 2015, ISBN 978-3-319-12771-2 , doi : 10.1007 / 978-3-319-12772-9 . }
• Richard Courant : Lectures on differential and integral calculus. First volume. Functions of a variable . Reprint 1948 (the 2nd edition from 1930). 2nd improved edition. Springer Verlag , Berlin 1948.
• GM Fichtenholz : Differential and Integral Calculus II . Translation from Russian and scientific editing: Dipl.-Math. Brigitte Mai, Dipl.-Math. Walter Mai (=  university books for mathematics . Volume 62 ). 6th edition. VEB Deutscher Verlag der Wissenschaften , Berlin 1974.
• Steven R. Finch: Mathematical Constants (=  Encyclopedia of Mathematics and its Applications . Volume 94 ). Cambridge University Press , Cambridge [u. a.] 2003, ISBN 0-521-81805-2 ( MR2003519 ).
• Otto Forster : Analysis 1. Differential and integral calculus of a variable (=  Vieweg Studium. Basic course in mathematics ). 9th, revised edition. Vieweg, Wiesbaden 2008, ISBN 978-3-8348-0395-5 .
• Hans Grauert , Ingo Lieb : Differential and integral calculus I. Functions of a real variable (=  Heidelberg pocket books . Volume 26 ). 4th improved edition. Springer Verlag, Berlin, New York 1976 ( MR0430171 ). }
• H. Jerome Keisler : Elementary Calculus: An Infinitesimal Approach . 3. Edition. Dover Publications, Mineola, NY 2012, ISBN 978-0-486-48452-5 .
• Konrad Knopp : Theory and application of the infinite series (=  The basic teachings of the mathematical sciences in individual presentations with special consideration of the areas of application . Volume 2 ). 5th, corrected edition. Springer Verlag, Berlin, Göttingen, Heidelberg, New York 1964, ISBN 3-540-03138-3 ( MR0183997 ).
• Herbert Meschkowski : Infinite rows . 2nd, improved and enlarged edition. BI Wissenschaftsverlag, Mannheim u. a. 1982, ISBN 3-411-01613-2 ( MR0671586 ).

Individual evidence

1. ↑ ^{a b c} Martin Barner, Friedrich Flohr: Analysis I . 5th edition. 2000, p. 145-146 .
2. ↑ ^{a b} Claudio Canuto, Anita Tabacco: Mathematical Analysis I . 2nd Edition. 2015, p. 151-152 .
3. ^ ^{A b c} Richard Courant: Lectures on differential and integral calculus. First volume . 2nd Edition. 1948, p. 295-298 .
4. ↑ GM Fichtenholz: Differential and Integral Calculus II . 6th edition. 1974, p. 315-317 .
5. ^ ^{A b c} Otto Forster: Analysis 1 . 9th edition. 2008, p. 66-68 .
6. ↑ Hans Grauert, Ingo Lieb: differential and integral calculus I. (Chapter III, definition 3.1) . 4th edition. 1976.
7. ↑ ^{a b c d e} I. N. Bronstein, KA Semendjajev u. a (Ed.): Pocket book of mathematics . 10th edition. 2016, p. 477-478 .
8. ↑ GM Fichtenholz: Differential and Integral Calculus II . 6th edition. 1974, p. 315-316 .
9. ^ ^{A b} Steven R. Finch: Mathematical Constants . 2003, p. 20 .
10. ^ Steven R. Finch: Mathematical Constants . 2003, p. 2 .
11. ^ Steven R. Finch: Mathematical Constants . 2003, p. 358 .
12. ^ ^{A b c} Steven R. Finch: Mathematical Constants . 2003, p. 43 .
13. ^ Steven R. Finch: Mathematical Constants . 2003, p. 53 .
14. ^ Steven R. Finch: Mathematical Constants . 2003, p. 434-436 .
15. ^ Steven R. Finch: Mathematical Constants . 2003, p. 167 .
16. ^ Steven R. Finch: Mathematical Constants . 2003, p. 96 .
17. ^ Steven R. Finch: Mathematical Constants . 2003, p. 34 .
18. ^ Steven R. Finch: Mathematical Constants . 2003, p. 449 .
19. ^ IN Bronstein, KA Semendjajev u. a (Ed.): Pocket book of mathematics . 10th edition. 2016, p. 1077 .
20. ^ Otto Forster: Analysis 1. 9th edition. 2008, p. 254-258 .
21. ^ Otto Forster: Analysis 1. 9th edition. 2008, p. 258 .
22. ^ Otto Forster: Analysis 1. 9th edition. 2008, p. 137-138, 253-254 .
23. ↑ GM Fichtenholz: Differential and Integral Calculus II . 6th edition. 1974, p. 317 .
24. ^ Steven R. Finch: Mathematical Constants . 2003, p. 53 .
25. ^ IN Bronstein, KA Semendjajev u. a (Ed.): Pocket book of mathematics . 10th edition. 2016, p. 576 .
26. ↑ H. Jerome Keisler: Elementary Calculus: An Infinitesimal Approach . 3. Edition. 2012, p. 520 .

Remarks

1. ↑ This criterion is named after Gottfried Wilhelm Leibniz . GM Fichtenholz denotes in his differential and integral calculus II - cf. there footnote on p. 315! - an alternating series, which meets the conditions of the Leibnizian criterion, as a series of the Leibnizian type .
2. ↑ Steven R. Finch also names the representation for the Apéry constant (cf. op. Cit. P. 43) . ${\displaystyle \zeta (3)=\sum _{k=2}^{\infty }{\sum _{m=1}^{k-1}{\frac {1}{k^{2}m}}}}$
3. ↑ This is the Sylvester episode named after James Joseph Sylvester . Compare with the article Sylvester's sequence available in the English language Wikipedia and episode A000058 in OEIS  !
4. ↑ According to Finch (cf. ibid. P. 436), the constants and are both transcendent numbers , while . Almost nothing is known so far (as of 2003) about the number . ${\displaystyle c}$ ${\displaystyle c^{'}}$ ${\displaystyle \sum _{k=0}^{\infty }{\frac {1}{s_{k}}}=1}$ ${\displaystyle \sum _{k=0}^{\infty }{\frac {1}{s_{k}-1}}=1{,}6910302067\ldots }$
5. ↑ According to Finch (cf. op. Cit. P. 43), the series representation also applies here . ${\displaystyle \sum _{k=2}^{\infty }{\frac {\zeta (k)-1}{k}}=1-\gamma }$
6. ↑ is nothing more than the logarithm of two . ${\displaystyle {\frac {\ln k}{\ln 2}}}$ ${\displaystyle k}$
7. ↑ Here we know from Euler's theorem that the series applies. Finch (cf. op. Cit., P. 96) further refers to the series that is also belonging to it , about which it is not yet known (as of 2003) whether it converges or diverges, which Paul Erdős formulated in 1996 as an open problem. ${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{p_{k}}}={\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{7}}+\ldots =\infty }$ ${\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k}\cdot k}{p_{k}}}=-{\frac {1}{2}}+{\frac {2}{3}}-{\frac {3}{5}}+{\frac {4}{7}}-+\ldots }$
8. ↑ Here Finch (... See a a O p 449) is improper for the associated integral also a number representation: . ${\displaystyle \int _{0}^{1}{\frac {1}{x^{x}}}\mathrm {d} x=1{,}2912859970\ldots }$ ${\displaystyle \int _{0}^{1}{\frac {1}{x^{x}}}\mathrm {d} x=\sum _{k=1}^{\infty }{\frac {1}{k^{k}}}=1+{\frac {1}{4}}+{\frac {1}{27}}+{\frac {1}{256}}+{\frac {1}{3125}}+\ldots }$
9. ↑ If so , you win the aforementioned example. ${\displaystyle x=1}$
10. ↑ In this case one wins the aforementioned Leibniz series. ${\displaystyle x=1}$
11. ↑ These Taylor series are absolutely convergent for even for all real numbers and also for all complex numbers . ${\displaystyle x}$
12. ↑ Finch (cf. op. Cit. P. 43) following, for example, the series expansion can be obtained from this . ${\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{\sqrt {k}}}={\frac {-1}{1+{\sqrt {2}}}}\cdot \lim _{n\to \infty }\left(1+{\frac {1}{\sqrt {2}}}+\ldots +{\frac {1}{\sqrt {n}}}-2\cdot {\sqrt {n}}\right)={\frac {1{,}4603545088\ldots }{1+{\sqrt {2}}}}=0{,}6048986434\ldots }$
13. ↑ Here one has . ${\displaystyle \beta (2)=G}$