Resolution of Grandi’s Paradox as Extended to Complex Valued Functions

Grandi’s paradox, which was posed for a real function of the form ( ) 1 1 x + , has been resolved and extended to complex valued functions. Resolution of this approximately three-hundred-year-old paradox is accomplished by the use of a consistent truncation approach that can be applied to all the series expansions of Grandi-type functions. Furthermore, a new technique for improving the convergence characteristics of power series with alternating signs is introduced. The technique works by successively averaging a series at different orders of truncation. A sound theoretical justification of the successive averaging method is demonstrated by two different series expansions of the function ( ) 1 1 e + . Grandi-type complex valued functions such as ( ) 1 i x + are expressed as consistently-truncated and convergence-improved forms and Fagnano’s formula is established from the series expansions of these functions. A Grandi-type general complex valued function ( ) 1 e s x θ ρ + is introduced and expanded to a consistently truncated and successively averaged series. Finally, an unorthodox application of the successive averaging method to polynomials is presented.

has been resolved and extended to complex valued functions. Resolution of this approximately three-hundred-year-old paradox is accomplished by the use of a consistent truncation approach that can be applied to all the series expansions of Grandi-type functions. Furthermore, a new technique for improving the convergence characteristics of power series with alternating signs is introduced. The technique works by successively averaging a series at different orders of truncation. A sound theoretical justification of the successive averaging method is demonstrated by two different series expansions of the function ( )

Introduction
Luigi Guido Grandi (1671-1742) is known due to his book entitled Quadratura in short. The book does not contain much original work except for two particular items; namely, the construction of a curve that has become known as the Witch of Agnesi and the identification of a paradox originating from the series expansion of ( ) 1 which obviously converges for 0 1 x ≤ ≤ and diverges for 1 x > . The upper limit of convergence, 1 x = , is named here as the threshold point for later purposes. Grandi remarked that for 1 x = the left-hand side of (1) would be ( ) Before proceeding to the resolution of the paradox the series expansion in (1) is rendered convergent for 1 x > by a simple manipulation as follows ( ) 1 1

Resolution of Grandi's Paradox
In tackling with Grandi's paradox the crucial point is to perceive the duality embedded in it. Starting from this recognition the paradox can be resolved as presented in detail in Beji [2]. Here, we shall first recapitulate the main aspects of that work and then proceed to extend it to complex valued functions.
The duality concerns the numerical value of series expansion depending on the number of terms included for the threshold value 1 x = . If one retains terms up to an odd power in the series, where n is an arbitrary integer, the resultant sum on the right is obviously 0 for 1 x = . On the other hand, if terms up to an even power are kept, setting 1 x = results in 1 on the right. Thus, depending on the highest power kept being odd or even the result on the right is either 0 or 1. In a sense, the series cannot be called precisely convergent for 1 x = . The paradox indeed pivots around this altercation of numerical values. The resolution must introduce a reconciliation such that taking just one more term should not result in any appreciable difference for 1 x = .  respectively 0 and 1 for the threshold point 1 x = but not the correct value 1/2. For this reason, series representations diverge rapidly either to 0 or 1 from the actual function ( ) (1) and (2)   proach may be followed when the highest power is even. The resulting expressions read: Accordingly, if the series is truncated by taking the one-half of the last term, the correct result 1/2 is obtained for the threshold point 1 x = , regardless of the truncation order. The resolution presented may at first sight appear as an ad hoc approach specifically devised to obtain the desired result; nevertheless, its soundness shall be more evident with further ramifications and different applications, as d'Alembert (1717-1783) said on the defence of calculus: "Allez en avant, et la foi vous viendra 1 ". 1 Go forward, and faith shall come.

A Convergence Improvement Technique
A technique for improving the convergence properties of truncated series is now introduced. The approach is demonstrated for the series expansion of ( )  Table 1 for a definite number of terms can be generalized to an arbitrary number of terms. Table 1. Successive averaging of the consistently truncated series expansions of ( ) For 1 x ≥ Equation (7) can be put into the following convergent form by the approach used in Equation (2): in which the truncation order n may be selected as an odd or even integer. The most striking feature of the series expansions (7) and (8)    Integration of (7) and (8) Note that when x → ∞ , ( ) ln 1 x + on the left grows boundlessly while the terms involving the powers of 1/x on the right go to zero. But ln x , arising from the integration of the first term 1/x on the right-side of Equation (8), takes care of this problem. An important point in using (9) and (10) together for computations is to dismiss the highest-order term 1 n x + in (9) completely to be consistent with the highest-order term 1 n x in (10). This crucial aspect is numerically

Mathematical Justification of Consistent Truncation and Convergence Improvement Techniques
In order to resolve Grandi's paradox in a suitable way, we have first introduced a consistent truncation approach and then a convergence improvement technique of successive averaging. While the satisfactory outcome of these procedures itself is a justification enough, relating both applications to a solid mathematical background is desirable. Grandi's function ( ) Carrying out a simple division gives After re-arranging the like-terms one obtains Equation (13) turns out to be which correctly satisfies 1 2 1 2 = when 0 x = . We are then facing an interesting case of two different approaches aimed at the same end but producing conflicting results. Obviously, Equation (14) is the correct result and if it can be obtained by employing the method applied to resolving Grandi's paradox we can claim a solid background for our approach. Thus, we begin with Equation (11) but apply the consistent truncation procedure followed by successive averaging as exactly done for the sample calculation given in §3, Collecting the terms of the same order together gives which, except for the fourth-order term, is essentially the same as Equation (14).
The disagreement in the fourth-order terms originates from the difference between the classical infinite series formulation and the truncated approach embedded in the present method. If a fifth-order expansion were carried out the fourth-order terms would be identical while the fifth-order terms are different.
Only for infinitely many terms would the present approach be identical with the classical one as the coefficient of the highest-order term tends to zero. This demonstration has thus revealed a subtle connection between the classical Maclaurin series expansion of a function and the consistent truncation and successive averaging technique introduced here to resolve Grandi's paradox. Such a far-fetched connection is unexpected but quite satisfying as it bolsters confidence in the method by providing firm theoretical support.

Grandi's Paradox for Complex Valued Functions
We now proceed to define complex valued functions which yield paradoxical results for definite x values when expanded into series just like ( ) where 1 i = − is the imaginary unit. Our primary aim here is to carry out a consistent truncation and convergence improvement of the series expansions of ( ) Before proceeding towards this goal it is appropriate to apply these methods to ( ) 2 1 1 x + so that the results can be used to establish the expansions of ( ) The standard series expansion of ( ) 2 1 1 x + is 2 4 6 8 10 2 1 1 1 x Obviously, the right-hand side of Equation (19) where n is an arbitrary truncation order.
At this stage, it is tempting to compute estimated π values by use of the standard and improved series expansions of ( ) . On the other hand, the right side at the upper limit would go to zero while at the lower limit would be multiplied by a minus sign, resulting in a correct positive estimate for π/4. Finally, of historic significance, it must be indicated that Equation (24) is originally due to Leibniz (1646-1716) [3], p. 10.
We now proceed to the treatment of ( ) , which corresponds to the real part. We must however observe that the real and imaginary parts have respectively odd and even powers of x and therefore truncation orders must be different for each part. Moreover, for exact result at the threshold point x i = the truncation power of the real part must be less than the imaginary part for 1 x ≤ and vice versa for 1 x ≥ . This arrangement of truncation orders has the additional advantage of giving exact functional value for 1 x = , just like a second threshold point. Bearing these precautions in mind we write down the consistently truncated and successively averaged series representation of ( ) ( ) ( ) The corresponding series for ( ) First, we set x i = + the threshold point in both (28)   1 would produce various paradoxical results for x i = + and 1 x = , depending on the truncation order.
Recalling that ( ) ln 1 i − = π and evaluating the right-hand side numerically, Equation (37) becomes 1 ln 1.571406372 1 Multiplying both sides by 2i and rearranging give 1 2 ln 3.140372563 1 in which the right-hand side is in 0.04% error compared to π. The absolute error is approximately π/2500 and the convergence rate of the series expansion with only 6 n = terms is excellent. Notice that the calculation of π in Equation (25) is actually 5 n = case of (37).

A General Complex Valued Function and Its Special Cases
where ρ and θ are real quantities defining a given complex constant, 1 s ≥ is a positive integer, and x is the independent variable which may be real, imaginary or complex. Obviously, Equations (40)     First, the terms multiplied by zero can be skipped; specifically, those multiplied S. Beji by cos 2 π , cos 2 3π , etc. and sin π , sin 2π , etc. as appear in (48) Right-hand sides of Equations (54) and (55)

An Unorthodox Application of Successive Averaging Method
Successive averaging method, which has been used for improving the convergence characteristics of the series is now applied to a polynomial to generate a family of polynomials with equal and lower orders that approximate the generic one to varying degrees in the neighbourhood of 0 x = . The method can be best explained by an example. Let us consider a fourthorder polynomial with known roots  Table 2 is exactly in line with Table 1; only now the terms making up the polynomial are used. We have then generated the family of four polynomials ( ) , which is 50% in error compared to the true root 1 x = ; hence not good at all. Nevertheless, the averaging approach may be used to obtain better approximations in a definite neighbourhood to a series representation of a given function as demonstrated in [2]; its further applications may be discovered in the future. The method begins with a consistent truncation followed by successive averaging for improving the convergence characteristics of the power series considered. Theoretical support for this methodology is provided via a demonstration using different series expansions of the function ( ) 1 1 e x + . Fagnano's formula is recovered in a non-trivial way by making use of the consistently-truncated and convergence-improved series expansions of ( ) In closing, an unorthodox use of the successive averaging method to polynomials is presented for suggesting diverse application areas.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.