Множење помоћу линија

Множење помоћу линија је метод веома популаран на интернету, такође познат под називима јапанско множење, кинеско множење, индијско множење, мајанско множење итсл. На сљедећем видеу можете видјети како се множе два броја овим методом. Мислим да видео све говори сам за себе и да додатно објашњење није потребно.

 

Провео сам мало истраживање и слиједећи изворе на интернету открио да је овај метод први пут постављен на интернет у видео облику 15.11.2006. Можете погледати оригиналан видео на сљедећем линку:

http://www.metacafe.com/watch/296904/easy_graphical_multiplication_trick/

Овај метод је веома користан при поучавању дјеце млађег узраста множењу природних (цијелих) бројева али није баш користан код множења великих бројева. Метод је базиран на дистрибутивности множења према сабирању:

21*13 = (2*10+1)*(1*10+3) = 2*1*100+(2*3*10+1*1*10)+1*3 = 273.

Поред свог уложеног времена и труда, нисам успио да пронађем када је се и гдје, овај метод први пут појавио. Врло је могуће да је овај забавни трик осмислио неко у новије вријеме да учини дјеци математику лакшом и занимљивијом и да је његово прво појављивање на интернету и уопште управо на претходном видео линку из 2006. године. То потврђује и један од коментара на сљедећи видео, са популарног youtube канала Vihart:

Screenshot_2017-12-05_22-12-53

У недостатку других извора, ако узмемо овај коментар и приче које круже интернетом за озбиљно, најзаслуженији назив за овај метод је – кинеско множење! 😀

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Top 10 Rated AI Movies

In this notebook, we will list the Top 10 rated AI movies based on their rating. We will modify and use Bayesian average method for calculating the rating for each movie with tag „artificial intelligence“ from average rating, number of user ratings and tag relevance score. Movies are rated and tagged by users of MovieLens website.

Jupyter Notebook 

This is a list of the top 10 rated movies tagged with „artificial intelligence“.

1 Matrix, The (1999) http://www.imdb.com/title/tt133093
2 Blade Runner (1982) http://www.imdb.com/title/tt83658
3 WALL·E (2008) http://www.imdb.com/title/tt910970
4 Moon (2009) http://www.imdb.com/title/tt1182345
5 2001: A Space Odyssey (1968) http://www.imdb.com/title/tt62622
6 Terminator 2: Judgment Day (1991) http://www.imdb.com/title/tt103064
7 Terminator, The (1984) http://www.imdb.com/title/tt88247
8 Ghost in the Shell (Kôkaku kidôtai) (1995) http://www.imdb.com/title/tt113568
9 Interstellar (2014) http://www.imdb.com/title/tt816692
10 Her (2013) http://www.imdb.com/title/tt1798709

This is a list of the top 10 rated AI movies that are not so famous and you (probably) have not seen them.

1. Her (2013)

2. Chappie (2015)

3. Automata (2014)

4. Colossus: The Forbin Project (1970)

5. The Machine (2013)

6. I, Robot (2004)

7. Transcendence (2014)

8. A.I. Artificial Intelligence (2001)

9. Short Circuit (1986)

10. Bicentennial Man (1999)

Andrew Ng: AI is the New Electricity

With so much information now online, a strong work ethic and growth mindset, even more than knowledge, predicts your future success.

– Andrew Ng –

Devin Coldewey talks to Andrew Ng about what makes an AI-first company, his vision for an AI-powered society, and the transformation of education.

Nick Bostrom vs Grady Booch (A.K.A. Elon Musk vs Mark Zuckerberg)

Nick Bostrom: What happens when ours computers get smarter than we are?

Artificial intelligence is getting smarter by leaps and bounds — within this century, research suggests, a computer AI could be as „smart“ as a human being. And then, says Nick Bostrom, it will overtake us: „Machine intelligence is the last invention that humanity will ever need to make.“ A philosopher and technologist, Bostrom asks us to think hard about the world we’re building right now, driven by thinking machines. Will our smart machines help to preserve humanity and our values — or will they have values of their own?

Ник Бостром: Шта ће се десити кад наши компјутери постану паметнији од нас?

Вештачка интелигенција напредује крупним корацима. У овом веку, кажу научници, компјутерска вештачка интелигенција могла би бити „паметна“ као човек. А онда ће нас, каже Ник Бостром, претећи: „Машинска интелигенција је последњи изум који ће човечанство морати да направи.“ Филозоф и технолог, Бостром нас моли да добро размислимо о свету који тренутно градимо и којим управљају мислеће машине. Да ли ће наше паметне машине помоћи да се очува човечанство и наше вредности – или ће имати своје сопствене вредности?

 

Grady Booch: Don’t fear superintelligent AI

New tech spawns new anxieties, says scientist and philosopher Grady Booch, but we don’t need to be afraid an all-powerful, unfeeling AI. Booch allays our worst (sci-fi induced) fears about superintelligent computers by explaining how we’ll teach, not program, them to share our human values. Rather than worry about an unlikely existential threat, he urges us to consider how artificial intelligence will enhance human life.

Грејди Буч: Не плашите се суперинтелигентне вештачке интелигенције

Нова технологија ствара нове неспокоје, каже научник и филозоф Грејди Буч, али не би требало да стрепимо од свемоћне, безосећајне вештачке интелигенције. Буч ублажава наше најгоре страхове (узроковане научном фантастиком) о суперинтелигентним компјутерима тако што нам објашњава како ћемо их подучавати, а не програмирати, да деле наше вредности. Уместо да се плашимо мало вероватне егзистенцијалне претње, тера нас да размислимо како ће вештачка интелигенција да унапреди људски живот.

Математика је скривена тајна за разумијевање света

Math is the hidden secret to understanding the world

Откључајте мистерије и унутрашње дјелање свијета кроз један од најмаштовитијих облика умјетности икада – математику – са Роџером Антонсеном, док он објашњава како мала промјена у перспективи може открити обрасце, бројеве и формуле као основ за емпатију и разумијевање. Видео је са сајта ted.com, доступан вам је српски пријевод.

Area of Torricelli’s Trumpet or Gabriel’s Horn, Sum of the Reciprocals of the Primes, Factorials of Negative Integers

Abstract. In our previous work [1], we defined the method for computing general limits of functions at their singular points and showed that it is useful for calculating divergent integrals, the sum of divergent series and values of functions in their singular points. In this paper, we have described that method and we will use it to calculate the area of Torricelli’s trumpet or Gabriel’s horn, the sum of the reciprocals of the primes and factorials of negative integers.

http://vixra.org/abs/1612.0413 [v1]

1. Introduction

Divergent series and divergent integrals have appeared in mathematics for a long time. Mathematicians have devised various means of assigning finite values to such series and integrals, although intuition suggests that the answer is infinity or it does not exist. Method for computing general limits of functions at their singular points, discovered in our previous work [1], will permit us to use the method of partial sums for calculating sums of divergent series and Newton – Leibniz formula for calculating divergent integrals, which is the new and surprising result. We also showed that our method is the strongest method around for summing divergent series and it is superior to other known methods; for more details we refer the reader to [2]. As for prerequisites, the reader is expected to be familiar with real and complex analysis in one variable.

In Section 2 we describe the method for computing general limits of functions at their singular points and show how that method may be used for assigning finite values to divergent series and divergent integrals. In this section, we present definitions and theorems with proofs because paper, [1] where the method is discovered, is not written in English.

In Sections 3, 4 and 5 we have compiled some of the standard facts on an area of Torricelli’s trumpet or Gabriel’s horn, the sum of the reciprocals of the primes and factorials of negative integers, respectively. In those sections, we assign finite values to an infinite area of Torricelli’s trumpet or Gabriel’s horn, the sum of divergent series of the reciprocals of the primes and Gamma function at their singular points, respectively. Gamma function extends factorials to real and even complex numbers. The gamma function is undefined for zero and negative integers, from which we can conclude that factorials of negative integers do not exist.

2. Method for Computing General Limits of Functions at Their Singular Points

Definition 2.1 Let {f} be a function and has a series expansion about the point {a \in {\mathbb C}\bigcup\infty}. We will denote by {\lim_{z\rightarrow a}^D f(z)} the general limit of function {f} at point {a} and define

\displaystyle lim_{z\rightarrow a}^D f(z)=c,

where {c} is constant term of any series expansion of {f} about {a}.

Example 2.1 The series expansions of {\sin z}, {\cos z} and {e^z} at infinity are same these functions and we considered that constant terms of their series expansions are {0}. By previous definition,

\displaystyle \lim_{z\rightarrow \infty}^D \sin z=\lim_{z\rightarrow \infty}^D \cos z=\lim_{z\rightarrow \infty}^D e^z=0.

Example 2.2 Let us find the general limit of Riemann zeta function as z approaches 1. The Laurent series expansions of a function {\zeta(z)} about {z=1} is the series {\frac{1}{z-1}+} {\gamma-} {\gamma _1(z-1)+} {\frac{1}{2}\gamma _2(z-1)^2-} {\frac{1}{6}\gamma_3(z-1)^3+} {\frac{1}{24}\gamma_4(z-1)^4+} {O((z-1)^5)}, where {\gamma} is Euler-Mascheroni constant and {\gamma_n} is the nth Stieltjes constant. By previous definition,

\displaystyle \lim_{z\rightarrow 1}^D \zeta(z)=\gamma.

Definition 2.2 Let {f} be a function and has a series expansion about the point {\infty}. We will denote by {\lim^D_{z\rightarrow\infty(\alpha)}f(z)}, {\lim^D_{z\rightarrow\infty(\alpha)^+}f(z)} and {\lim^D_{z\rightarrow\infty(\alpha)^-}f(z)} the general limit, upper general limit and lower general limit of a function f(z) as z approaches to {\infty} over radial line {l_{\alpha, \infty}=\{r\cdot e^{i\alpha}|r\in {\mathbb R}\}}, {\alpha\in[0,2\pi)}, respectively, and define

\displaystyle \lim^D_{z\rightarrow\infty(\alpha)^+}f(z)=\lim^D_{z\rightarrow+e^{i\alpha}\infty}f(z)=

\displaystyle \lim^D_{r\rightarrow\infty(0)^+}f(r\cdot e^{i\alpha})=\lim^D_{r\rightarrow +\infty}f(r\cdot e^{i\alpha}),

\displaystyle \lim^D_{z\rightarrow\infty(\alpha)^-}f(z)=\lim^D_{z\rightarrow-e^{i\alpha}\infty}f(z)=

\displaystyle \lim^D_{r\rightarrow\infty(0)^-}f(r\cdot e^{i\alpha})=\lim^D_{r\rightarrow -\infty}f(r\cdot e^{i\alpha}),

\displaystyle \lim^D_{z\rightarrow\infty(\alpha)}f(z)=\frac{1}{2}(\lim^D_{z\rightarrow\infty(\alpha)^+}f(z)+\lim^D_{z\rightarrow\infty(\alpha)^-}f(z))=

\displaystyle \lim^D_{r\rightarrow\infty(0)}f(r\cdot e^{i\alpha}).

 

Definition 2.3 Let {f} be a function and has a series expansion about the point {a\in{\mathbb C}}. We will denote by {\lim^D_{z\rightarrow a(\alpha)}f(z)}, {\lim^D_{z\rightarrow a(\alpha)^+}f(z)} and {\lim^D_{z\rightarrow a(\alpha)^-}f(z)} the general limit, upper general limit and lower general limit of a function f(z) as z approaches to {a} over radial line {l_{\alpha,a}=\{a+r\cdot e^{i\alpha}|r\in {\mathbb R}\}}, {\alpha\in[0,2\pi)}, respectively, and define

\displaystyle \lim^D_{z\rightarrow a(\alpha)^+}f(z)=\lim^D_{z\rightarrow a+e^{i\alpha 0}}f(z)=

\displaystyle \lim^D_{r\rightarrow 0(0)^+}f(a+r\cdot e^{i\alpha})=\lim^D_{r\rightarrow 0^+}f(a+r\cdot e^{i\alpha}),

\displaystyle \lim^D_{z\rightarrow a(\alpha)^-}f(z)=\lim^D_{z\rightarrow a-e^{i\alpha 0}}f(z)=

\displaystyle \lim^D_{r\rightarrow 0(0)^-}f(a+r\cdot e^{i\alpha})=\lim^D_{r\rightarrow 0^-}f(a+r\cdot e^{i\alpha}),

\displaystyle \lim^D_{z\rightarrow a(\alpha)}f(z)=\frac{1}{2}(\lim^D_{z\rightarrow a(\alpha)^+}f(z)+\lim^D_{z\rightarrow a(\alpha)^-}f(z))=

\displaystyle \lim^D_{r\rightarrow 0(0)}f(a+r\cdot e^{i\alpha}).

 

Definition 2.4 Let {f} be a function and has a pole of order {m\in {\mathbb N}} at {a \in {\mathbb C}\bigcup\infty}. Define

\displaystyle lim_{z\rightarrow +\infty}^D P_n(z)=\int_{-1}^0 P_n(z) dz,

\displaystyle lim_{z\rightarrow -\infty}^D P_n(z)=\int_0^1 P_n(z) dz,

\displaystyle lim_{z\rightarrow \infty(0)}^D P_n(z)=\frac{1}{2}\int_{-1}^1 P_n(z) dz,

\displaystyle lim_{z\rightarrow 0^{+}}^D P_n\Big(\frac{1}{z}\Big)=\int_{-\infty}^{-1} P_n\Big(\frac{1}{z}\Big)\cdot\frac{1}{z^2} dz,

\displaystyle lim_{z\rightarrow 0^{-}}^D P_n\Big(\frac{1}{z}\Big)=\int_1^{+\infty} P_n\Big(\frac{1}{z}\Big)\cdot\frac{1}{z^2} dz,

\displaystyle lim_{z\rightarrow 0(0)}^D P_n\Big(\frac{1}{z}\Big)=\frac{1}{2}\int_{1}^{-1} P_n\Big(\frac{1}{z}\Big)\cdot\frac{1}{z^2} dz,

\displaystyle \lim_{z\rightarrow a(\alpha)}^{D}f(z)=\lim_{z\rightarrow a(\alpha)}^{D} F_1(z)+c_0,

where {P_n(z)=a_nz^n+a_{n-1}z^{n-1}+\cdot \cdot \cdot +a_1z+a_0} is polynomial of degree {n\geq 0}, {c_0} is a constant term and {F_1(z)} is the principal part of a Laurent series expansion of {f} at {a}.

Example 2.3 Let us find the general limit of a Riemann zeta function as z approaches 1 over radial line {l_{0, 1}}, where {l_{0, 1}} is real axis. The Laurent series expansions of a function {\zeta(z)} about {z=1} is the series {\frac{1}{z-1}+} {\gamma-} {\gamma_1(z-1)+} {\frac{1}{2}\gamma_2(z-1)^2-} {\frac{1}{6}\gamma_3(z-1)^3+} {\frac{1}{24}\gamma _4(z-1)^4+} {O((z-1)^5)}, where {\gamma} is Euler-Mascheroni constant and {\gamma_n} is the nth Stieltjes constant. By previous definitions,

\displaystyle \lim_{z\rightarrow 1(0)}^D \zeta(z)=\lim_{z\rightarrow 1(0)}^D\frac{1}{z-1}+\gamma=\lim_{r\rightarrow 0(0)}^D\frac{1}{1+r-1}+\gamma=

\displaystyle \frac{1}{2}\int_{1}^{-1} \frac{1}{r}\cdot\frac{1}{r^2} dr+\gamma=0+\gamma=\gamma.

Example 2.4 Let us find the sum of divergent series {\sum_{n=1}^{\infty}1=1+1+1+1+\cdot\cdot\cdot +1+\cdot\cdot\cdot}. Thus, by previous definition,

\displaystyle \sum_{n=1}^{\infty}1=\lim^D_{m\rightarrow+\infty}\sum_{n=1}^m 1=\lim^D_{m\rightarrow+\infty}m=\int_{-1}^0 m dm=\frac{-1}{2}.

Example 2.5 Let us find the sum of divergent series {\sum_{n=1}^{\infty}n^k=1^k+2^k+3^k+4^k+\cdot\cdot\cdot +m^k+\cdot\cdot\cdot}, where {k} is positive integer. By Faulhaber’s formula, {\sum_{n=1}^{m}n^k=\frac{1}{k+1}\sum_{n=0}^k (-1)^n{{k+1}\choose{n}} B_n m^{k+1-n}} since {B_1=-\frac{1}{2}}, where {B_n} denotes the nth Bernoulli number. Therefore {\sum_{n=1}^{\infty}n^k= \lim^D_{m\rightarrow+\infty}\sum_{n=1}^m n^k=} {\lim^D_{m\rightarrow+\infty}(\frac{1}{k+1}\sum_{n=0}^k (-1)^n{{k+1}\choose{n}} B_n m^{k+1-n})=} {\int_{-1}^0(\frac{1}{k+1}\sum_{n=0}^k (-1)^n{{k+1}\choose{n}} B_n m^{k+1-n})dm=} {\frac{1}{k+1}\sum_{n=0}^k (-1)^n{{k+1}\choose{n}} B_n \int_{-1}^0 m^{k+1-n}dm=} {-\frac{1}{k+1}\sum_{n=0}^k (-1)^n{{k+1}\choose{n}} B_n \frac{(-1)^{k+2-n}}{k+2-n}=} {-\frac{1}{k+1}\sum_{n=0}^k {{k+1}\choose{n}} B_n \frac{(-1)^{k}}{k+2-n}=} {\frac{(-1)^{k}}{k+1}\cdot(-\sum_{n=0}^k {{k+1}\choose{n}} \frac{B_n}{k+2-n})=} {\frac{(-1)^{k}}{k+1}\cdot B_{k+1}} by recurrence equation for Bernoulli numbers and previous definition. We have

\displaystyle \sum_{n=1}^{\infty}n^k=-\frac{B_{k+1}}{k+1}

since {k\in {\mathbb N}}, because the odd Bernoulli numbers are zero.

Theorem 2.1 If {f} is a function and has a pole of order 1 at {a \in {\mathbb C}\bigcup\infty} and if {c_0} is a constant term of a Laurent series expansion of {f} at {a}, then

\displaystyle \lim^D_{z\rightarrow a(\alpha)}f(z)=c_0,\ \alpha\in[0,2\pi).

Proof: By previous definition, {\lim^D_{z\rightarrow a(\alpha)}f(z)=} {\lim^D_{z\rightarrow a(\alpha)}F_1(z)+c_0=} {\lim^D_{z\rightarrow a(\alpha)}\frac{c_{-1}}{z-a}+c_0=} {\lim^D_{r\rightarrow 0(0)}\frac{c_{-1}}{a+re^{i\alpha}-a}+c_0=} {\frac{1}{2}\int_1^{-1}\frac{c_{-1}}{re^{i\alpha}}\cdot\frac{1}{r^2}dr+c_0=} {\frac{c_{-1}}{2e^{i\alpha}}\int_1^{-1}\frac{1}{r^3}dr+c_0}. Similarly we can prove that the theorem holds for {a=\infty}. \Box

Example 2.6 Let us find the general limit of a Gamma function, denoted by {\Gamma(z)}, as z approaches 0 over radial line {l_{0, 0}}, where {l_{0, 0}} is real axis. The Laurent series expansions of a function {\Gamma(z)} about {z=0} is the series {\frac{1}{z}-} {\gamma+} {\frac{1}{12}(6\gamma^2+\pi^2)z+} {\frac{1}{6}z^2(-\gamma^3-\frac{\gamma\pi^2}{2}+} {\mit\psi^{(2)}(1))+} {\frac{1}{24}z^3(\gamma^4+\gamma^2\pi^2+\frac{3\pi^4}{20}-4\gamma\mit\psi^{(2)}(1))+} {\frac{1}{1440}z^4(-12\gamma^5-20\gamma^3\pi^2-9\gamma\pi^4+120\gamma^2\mit\psi^{(2)}(1)+} {20\pi^2\mit\psi^{(2)}(1)+12\mit\psi^{(4)}(1))+O(z^5)}, where {\gamma} is Euler-Mascheroni constant and {\mit\psi^{(2)}(z)} is the nth derivative of the digamma function. By previous theorem,

\displaystyle \lim_{z\rightarrow 0(0)}^D \Gamma(z)=\lim_{z\rightarrow 0}^D \Gamma(z)=-\gamma.

Definition 2.5 Let {f} be a function and has a series expansion about the point {a \in {\mathbb C}\bigcup\infty} and does not have a pole at {a}.

\displaystyle \lim^D_{z\rightarrow a(\alpha)^+}f(z)=c\ (\lim^D_{z\rightarrow a(\alpha)^-}f(z)=c)

if {\lim_{z\rightarrow a(\alpha)^+}f(z)} {(\lim_{z\rightarrow a(\alpha)^-}f(z))} is infinite or does not exist, where {c} is a constant term of any series expansion of {f} about {a}; otherwise

\displaystyle \lim^D_{z\rightarrow a(\alpha)^+}f(z)=\lim_{z\rightarrow a(\alpha)^+}f(z)

\displaystyle (\lim^D_{z\rightarrow a(\alpha)^-}f(z)=\lim_{z\rightarrow a(\alpha)^-}f(z)).

Example 2.7 Let us find the sum of the harmonic series which are divergent. We have {\sum_{n=1}^{\infty}\frac{1}{n}=\lim^D_{m\rightarrow+\infty}\sum_{n=1}^m \frac{1}{n}=\lim^D_{m\rightarrow+\infty}H_m}, where {H_m} is harmonic number. Therefore, by previous definition,

\displaystyle \sum_{n=1}^{\infty}\frac{1}{n}=\gamma,

where {\gamma} is Euler-Mascheroni constant, because the series expansions of a function {H_m} about {m=\infty} is the series {(\gamma-\ln(\frac{1}{m}))+} {\frac{1}{2m}-} {\frac{1}{12m^2}+} {\frac{1}{120m^4}-} {\frac{1}{252m^6}+} {O((\frac{1}{m})^7)}, where {\ln(z)} is natural logarithm.

Example 2.8 Let us find the sum of divergent series {\sum_{n=1}^{\infty}(n-1)!=0!+1!+2!+3!+\cdot\cdot\cdot +(n-1)!+\cdot\cdot\cdot}. We have {\sum_{n=1}^{\infty}(n-1)!=} {\lim^D_{m\rightarrow+\infty}\sum_{n=1}^m (n-1)!=} {\lim^D_{m\rightarrow+\infty}(-1)^m m!!(-m-1)+!(-2)+1}, where {n!!} is the double factorial function and {!n} is subfactorial function. Therefore, by previous definition,

\displaystyle \sum_{n=1}^{\infty}(n-1)!\approx 0.697175 + 1.15573 \cdot i,

because the constant term of a series expansion of function {(-1)^m m!!(-m-1)+!(-2)+1} about {\infty} are {1+\frac{\Gamma(-1,-1)}{e}=} {0.69717488323506606876547868191955159531717543095436951732...+} {1.1557273497909217179100931833126962991208510231644158204... \cdot i}, where {\Gamma(a,z)} is the incomplete gamma function.

Example 2.9 Let us find the finite value of divergent integral {\int_0^{+\infty} \sin x dx}. We have {\int_0^{+\infty} \sin x dx=} {(-\cos x)|_0^{+\infty}=} {\lim^D_{x\rightarrow+\infty}(-\cos x)+\cos 0}. Thus, by previous definition,

\displaystyle \int_0^{+\infty} \sin x dx=0+1=1,

because the series expansions of {-\cos x} at {\infty} are {-\cos x=-\cos x+0}.

Example 2.10 Let us find the finite value of divergent integral {\int_0^{+\infty} \ln x \sin x dx}, where {\ln x} is natural logarithm. We have {\int_0^{+\infty} \ln x \sin x dx=} {(Ci(x)-\ln x \cos x) |_0^{+\infty}=} {\lim_{x\rightarrow+\infty}^D(Ci(x)-\ln x \cos x)-} {\lim_{x\rightarrow 0}^D(Ci(x)-\ln x \cos x)=} {0-\gamma=} {-\gamma}, where {Ci(x)} is cosine integral and {\gamma} is Euler-Mascheroni constant. Thus, by previous definition,

\displaystyle \int_0^{+\infty} \ln x \sin x dx=0-\gamma=-\gamma,

because the series expansions of {Ci(x)-\ln x\cos x} at {\infty} are {\cos x(\ln\frac{1}{x}+} {O((\frac{1}{x})^7))+} {\cos x(-(\frac{1}{x})^2+} {\frac{6}{x^4}-} {\frac{120}{x^6}+} {O((\frac{1}{x})^7))+} {\sin x(\frac{1}{x}-} {\frac{2}{x^3}+} {\frac{24}{x^5}+} {O((\frac{1}{x})^7))+} {O((\frac{1}{x})^9)-} {i\pi\lfloor\frac{1}{2}-} {\frac{arg(x)}{\pi}\rfloor+0} and the series expansions of {Ci(x)-\ln x\cos x} at {0} are {\gamma+} {\frac{1}{4}x^2(2\ln x-1)+} {\frac{1}{96}x^4(1-4\ln x)+} {\frac{1}{4320}x^6(6\ln x-1)+} {O(x^7)}.

Theorem 2.2 If {f} is a function and has a pole of order {m\in {\mathbb N}} at {a \in {\mathbb C}\bigcup\infty} and if {c_0} is a constant term of a Laurent series expansion of {f} at {a}, then {lim_{z\rightarrow a}^D f(z)} is a mean value of general limits {\lim^D_{z\rightarrow a(\alpha)}f(z)}, {\alpha\in[0,2\pi)}.

Proof: Let us first prove that the theorem holds for {a \in {\mathbb C}}. By previous definitions and the first mean value theorem for definite integrals, {\lim^D_{z\rightarrow a}f(z)=} {\frac{1}{2\pi}\cdot\int_0^{2\pi}\lim^D_{z\rightarrow a(\alpha)}f(z)d\alpha=} {\frac{1}{2\pi}\cdot\int_0^{2\pi} \lim^D_{z\rightarrow a(\alpha)}F_1(z)d\alpha+c_0=} {\frac{1}{2\pi}\cdot\int_0^{2\pi} \frac{1}{2}(\lim^D_{z\rightarrow a(\alpha)^+}F_1(z)+\lim^D_{z\rightarrow a(\alpha)^-}F_1(z))d\alpha+c_0=} {\frac{1}{2\pi}\cdot\int_0^{2\pi} \frac{1}{2}(\lim^D_{r\rightarrow 0^+}F_1(a+re^{i\alpha})+\lim^D_{z\rightarrow a(\alpha)^-}F_1(a+re^{i\alpha}))d\alpha+c_0=} {\frac{1}{2\pi}\cdot\int_0^{2\pi} \frac{1}{2}(\lim^D_{r\rightarrow 0^+}\sum_{k=-m}^{-1}c_k(a+re^{i\alpha}-a)^k+\lim^D_{r\rightarrow 0^+}\sum_{k=-m}^{-1}c_k(a+re^{i\alpha}-a)^k)d\alpha+c_0=} {\frac{1}{2\pi}\cdot\int_0^{2\pi} \frac{1}{2}(\int_{-\infty}^{-1}\sum_{k=-m}^{-1}c_k(re^{i\alpha})^kdr+\int_1^{+\infty}\sum_{k=-m}^{-1}c_k(re^{i\alpha})^kdr)d\alpha+c_0=} {\frac{1}{2\pi}\cdot\int_0^{2\pi}\frac{1}{2}\sum_{k=-n}^{-1} c_k \frac{1+(-1)^k}{-k+1} e^{i\alpha k}d\alpha+c_0=} {\frac{1}{2\pi}\cdot\frac{1}{2}\sum_{k=-n}^{-1} c_k \frac{1+(-1)^k}{-k+1} \int_0^{2\pi}e^{i\alpha k}d\alpha+c_0=} {0+c_0=c_0}, where {F_1(z)} is the principal part of a Laurent series expansion of {f} at {a}. Similarly we can prove that the theorem holds for {a=\infty}. \Box

3. Area of Torricelli’s Trumpet or Gabriel’s Horn

Torricelli’s Trumpet, also called Gabriel’s Horn, a mathematical figure that stretched to infinity but was not infinitely big is the surface of revolution obtained by rotating the graph of the function {f(x)=\frac{1}{x}} on the interval {[1,\infty)} around the {x}-axis. Using integration, it is possible to find the surface area A:

\displaystyle A=2\pi\int_1^{+\infty}\frac{1}{x}\cdot\sqrt{1+\Big[\Big(\frac{1}{x}\Big)'\Big]^2}dx=2\pi\int_1^{+\infty}\frac{1}{x}\cdot\sqrt{1+\frac{1}{x^4}}dx\geq

\displaystyle 2\pi\int_1^{+\infty}\frac{1}{x}dx=+\infty.

Let us find the finite value of divergent integral {A=2\pi\int_1^{+\infty}\frac{1}{x}\cdot\sqrt{1+\frac{1}{x^4}}dx}. We have {2\pi\int_1^{+\infty}\frac{1}{x}\cdot\sqrt{1+\frac{1}{x^4}}dx=} {2\pi\left(\frac{\sqrt{\frac{1}{x^4}+1}x^2\sinh ^{-1}\left(x^2\right)}{2\sqrt{x^4+1}}-\frac{1}{2}\sqrt{\frac{1}{x^4}+1}\right)\Big|_1^{+\infty}.} Thus, {\lim^D_{x\rightarrow +\infty}2\pi\Big(\frac{\sqrt{\frac{1}{x^4}+1}x^2\sinh^{-1}(x^2)}{2\sqrt{x^4+1}}-\frac{1}{2} \sqrt{\frac{1}{x^4}+1}\Big)-} {\lim_{x\rightarrow 1}2\pi\Big(\frac{\sqrt{\frac{1}{x^4}+1}x^2\sinh^{-1}(x^2)}{2\sqrt{x^4+1}}-\frac{1}{2}\sqrt{\frac{1}{x^4}+1}\Big)=} {\frac{1}{2}\pi\ln(4)-\pi-2\pi(\frac{1}{2}\sinh ^{-1}(1)-\frac{1}{\sqrt{2}})\approx 0.70996}, where {\ln(z)} is natural logarithm and {\sinh^{-1}(z)} is the inverse hyperbolic sine function, because the series expansions of {2 \pi \Big(\frac{\sqrt{\frac{1}{x^4}+1}x^2\sinh^{-1}(x^2)}{2\sqrt{x^4+1}}-\frac{1}{2} \sqrt{\frac{1}{x^4}+1}\Big)} at {\infty} are {(-2\pi\ln(\frac{1}{x})-\pi+\frac{1}{2}\pi\ln(4))-\frac{\pi}{4x^4}+\frac{\pi}{32x^8}+O((\frac{1}{x})^{11})}. This gives

\displaystyle A=0.70995...

.

4. Sum of the Reciprocals of the Primes

The sum of the reciprocals of all prime numbers diverges. This was proved by Leonhard Euler in 1737, and strengthens Euclid’s 3rd-century-BC result that there are infinitely many prime numbers. We will denote by {p_n} nth prime number. Let us find the sum of series {\sum_{n=1}^{+\infty}\frac{1}{p_n}}. We have {\sum_{n=1}^{\infty}\frac{1}{p_n}=\lim^D_{m\rightarrow+\infty}\sum_{n=1}^m \frac{1}{p_n}=\lim^D_{s\rightarrow 1}P(s)}, where {P(s)\equiv\displaystyle\sum_{p\ is\ prime}\frac{1}{p^s}} is the prime zeta function. For {s} close to 1, {P(s)} has the expansion {P(1+\epsilon)=-\ln\epsilon+C+O(\epsilon)}, where {\epsilon\geq 0} and {C=\sum_{n=2}^{+\infty} \frac{\mu(n)}{n}\ln\zeta(n)=} {M-\gamma=0.261497212...-0.577215664...=} {-0.315718452...}, where {M} is Meissel-Mertens constant, {\gamma} is Euler-Mascheroni constant, {\mu(n)} is the M\“{o}bius function, {\zeta(n)} is the Riemann zeta function and {\ln(z)} is natural logarithm. Therefore, {\sum_{n=1}^{+\infty}\frac{1}{p_n}=\lim^D_{\epsilon\rightarrow 0}P(1+\epsilon)=\lim^D_{\epsilon\rightarrow 0}(-\ln\epsilon+C+O(\epsilon))=C} because the series expansions of a function {-\ln\epsilon+C+O(\epsilon)} about {\epsilon=0} is the series {(C+\ln (\epsilon))+O\left(\epsilon^1\right)}. This gives

\displaystyle \sum_{n=1}^{+\infty}\frac{1}{p_n}=M-\gamma=-0.315718452...

.

5. Factorials of Negative Integers

The gamma function was first introduced by Leonhard Euler in his goal to generalize the factorial to non integer values. The (complete) gamma function {\Gamma(z)=\int_0^{+\infty}x^{z-1}e^{-x}dx} is defined to be an extension of the factorial to complex and real number arguments. It is analytic everywhere except at {z=0, -1, -2, ...}, where it has a poles of order 1. It is related to the factorial by {\Gamma(n+1)=(n)!} as special case of functional equation {\Gamma(z+1)=z\Gamma(z)}. Gamma function is not the only solution of the previous functional equation. Let us find the factorials of negative numbers as the general limit of a gamma function as {z} approaches {-n}, where {n} are positive integers. We have {(-n)!=\lim^D_{z\rightarrow -n}\Gamma(z)=c(n)}, where {c(n)} denote the constant term of the Laurent series expansion of a function {\Gamma(z)} about {z=-n}. This gives

\displaystyle (-1)!=-1+\gamma,

\displaystyle (-2)!=\frac{3}{4}-\frac{\gamma}{2},

\displaystyle (-3)!=-\frac{11}{36}+\frac{\gamma}{6},

\displaystyle (-4)!=\frac{25}{288}-\frac{\gamma}{24},

…, where {\gamma} is Euler-Mascheroni constant.

References

[1] Sinisa Bubonja, General Method for Summing Divergent Series. Determination of Limits of Divergent Sequences and Functions in Singular Points, Preprint, viXra:1502.0074

[2] Sinisa Bubonja, General Method for Summing Divergent Series Using Mathematica and a Comparison to Other Summation Methods, Preprint, viXra:1511.0247

[3] G. H. Hardy, Divergent series, Oxford at the Clarendon Press (1949)

[4] Bruce C. Brendt, Ramanujan’s Notebooks, Springer-Verlag New York Inc. (1985)

[5] John Tucciarone, The Development of the Theory of Summable Divergent Series from 1880 to 1925, Archive for History of Exact Sciences, Vol. 10, No. 1/2, (28.VI.1973), 1-40