**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 2.1Let be a function and has a series expansion about the point . We will denote by the general limit of function at point and define

where is constant term of any series expansion of about .

Example 2.1The series expansions of , and at infinity are same these functions and we considered that constant terms of their series expansions are . By previous definition,

Example 2.2Let us find the general limit of Riemann zeta function as z approaches 1. The Laurent series expansions of a function about is the series , where is Euler-Mascheroni constant and is the nth Stieltjes constant. By previous definition,

Definition 2.2Let be a function and has a series expansion about the point . We will denote by , and the general limit, upper general limit and lower general limit of a function f(z) as z approaches to over radial line , , respectively, and define

Definition 2.3Let be a function and has a series expansion about the point . We will denote by , and the general limit, upper general limit and lower general limit of a function f(z) as z approaches to over radial line , , respectively, and define

Definition 2.4Let be a function and has a pole of order at . Define

where is polynomial of degree , is a constant term and is the principal part of a Laurent series expansion of at .

Example 2.3Let us find the general limit of a Riemann zeta function as z approaches 1 over radial line , where is real axis. The Laurent series expansions of a function about is the series , where is Euler-Mascheroni constant and is the nth Stieltjes constant. By previous definitions,

Example 2.4Let us find the sum of divergent series . Thus, by previous definition,

Example 2.5Let us find the sum of divergent series , where is positive integer. By Faulhaber’s formula, since , where denotes the nth Bernoulli number. Therefore by recurrence equation for Bernoulli numbers and previous definition. We have

since , because the odd Bernoulli numbers are zero.

Theorem 2.1If is a function and has a pole of order 1 at and if is a constant term of a Laurent series expansion of at , then

*Proof:* By previous definition, . Similarly we can prove that the theorem holds for .

Example 2.6Let us find the general limit of a Gamma function, denoted by , as z approaches 0 over radial line , where is real axis. The Laurent series expansions of a function about is the series , where is Euler-Mascheroni constant and is the nth derivative of the digamma function. By previous theorem,

Definition 2.5Let be a function and has a series expansion about the point and does not have a pole at .

if is infinite or does not exist, where is a constant term of any series expansion of about ; otherwise

Example 2.7Let us find the sum of the harmonic series which are divergent. We have , where is harmonic number. Therefore, by previous definition,

where is Euler-Mascheroni constant, because the series expansions of a function about is the series , where is natural logarithm.

Example 2.8Let us find the sum of divergent series . We have , where is the double factorial function and is subfactorial function. Therefore, by previous definition,

because the constant term of a series expansion of function about are , where is the incomplete gamma function.

Example 2.9Let us find the finite value of divergent integral . We have . Thus, by previous definition,

because the series expansions of at are .

Example 2.10Let us find the finite value of divergent integral , where is natural logarithm. We have , where is cosine integral and is Euler-Mascheroni constant. Thus, by previous definition,

because the series expansions of at are and the series expansions of at are .

Theorem 2.2If is a function and has a pole of order at and if is a constant term of a Laurent series expansion of at , then is a mean value of general limits , .

*Proof:* Let us first prove that the theorem holds for . By previous definitions and the first mean value theorem for definite integrals, , where is the principal part of a Laurent series expansion of at . Similarly we can prove that the theorem holds for .

**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 on the interval around the -axis. Using integration, it is possible to find the surface area A:

Let us find the finite value of divergent integral . We have Thus, , where is natural logarithm and is the inverse hyperbolic sine function, because the series expansions of at are . This gives

.

**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 nth prime number. Let us find the sum of series . We have , where is the prime zeta function. For close to 1, has the expansion , where and , where is Meissel-Mertens constant, is Euler-Mascheroni constant, is the M\“{o}bius function, is the Riemann zeta function and is natural logarithm. Therefore, because the series expansions of a function about is the series . This gives

.

**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 is defined to be an extension of the factorial to complex and real number arguments. It is analytic everywhere except at , where it has a poles of order 1. It is related to the factorial by as special case of functional equation . 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 approaches , where are positive integers. We have , where denote the constant term of the Laurent series expansion of a function about . This gives

…, where 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

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