Banishing divergence Part 2: Limits of oscillatory sequences, and applications
BBanishing divergence Part 2: Limits ofoscillatory sequences, and applications.
David Alan PatersonCSIRO CMSEGraham Rd, Highett, 3090AustraliaOctober 9, 2018
Abstract
Sequences diverge either because they head off to infinity or be-cause they oscillate. Part 1 [1] of this paper laid the pure mathematicsgroundwork by defining Archimedean classes of infinite numbers aslimits of smooth sequences. Part 2 follows that with applied mathe-matics, showing that general sequences can usually be converted intosmooth sequences, and thus have a well-defined limit.Each general sequence is split into the sum of smooth, periodic (in-cluding Lebesgue integrable), chaotic and random components. Themean of each of these components divided by a smooth sequence, orthe mean of the mean, will usually be a smooth sequence, and so theoscillatory sequence will have at least a leading term limit. Examplesillustrate the wide range of oscillatory sequences that have leadingterm limits.Methologies are given for applications of such limits in four othercontexts. One is for finding the limit of a misbehaving function ata point on the real number line. A second uses a nonstandard typeof contour integration to find the limit of a function on the complexnumbers. The third application is to Riemann sums for evaluatingimproper integrals. The final application is in evaluating the squareof the Dirac delta function.
This paper is the first step towards a mathematical formalism in which everyinfinite sequence of real numbers has a limit. Part 1 [1] defined infinite and1 a r X i v : . [ m a t h . G M ] A ug nfinitesimal numbers in a new way, as limits of a specific ”smooth” subsetof sequences.Part 2 handles handles limits of general sequences and in particular oscil-latory sequences lim n → ω f ( n ). Each sequence is split into the sum of smooth s , periodic p (including Lebesgue integrable), chaotic k and random r com-ponents as follows: f ( n ) = s ( n )+ (cid:80) j j s ( n ) j p ( n )+ (cid:80) k k s ( n ) k k ( n )+ (cid:80) l l r ( n )where each appearance of s can be a different smooth sequence. The mean ofeach of these components, or the mean of the mean, will usually be a smoothsequence. Care is taken to ensure that each mean is definied locally at asufficiently large n rather than being smeared over several values of n .This use of the mean maps general sequences onto smooth sequences and,since smooth sequences have a limit by Part 1 of this paper, many if not allsequences have a limit. Examples of difficult sequences with well-definedlimits are presented.This paper then presents methologies for applications of such limits onthe real x, a and complex z, z numbers lim x → a f ( x ), lim z → z f ( z ), for anapplication to improper integrals as Riemann sums (cid:82) ω f ( x ) dx = lim n → ω √ n (cid:80) nk =1 f (( k − . / √ n ) and similar, and the paper finishes with a brief finalnote about squaring the Dirac delta function.All of the work presented in this paper requires only elementary mathe-matics. All the techniques would have been accessible to any mathematicianliving 100 years ago, and indeed five separate pivotal papers on which Part 1rests were published between 1895 and 1907. If this work hasn’t appeared inprint before, then the only reason I can think of is that this paper touches verybriefly on a remarkably broad range of mathematical techniques, includinginfinite ordinal numbers, big O notation, axioms of set theory, Box-Jenkinstime series analysis, probability and the ensemble mean, strange attractors,real and complex analysis and generalized functions.As much as possible of the work presented here is new. The following is a summary of the notation defined in Part 1 [1] that isneeded by Part 2.
Finite commutativity resolves many classical paradoxes involving infinityby rejecting the process on bijection [2] on ordered sets. ω is the first infinite ordinal number. What matters for this paper isthat ω is bigger than all integers and that it’s defined to be commutative: ω (cid:54) = 1 + ω = ω + 1. A typical infinitesimal number is 1 /ω .2n infinite sequence of real numbers is a map s from the natural numbersto the real numbers. Write as s ( n ) or s n .A prototype is a special type of sequence, denoted p n or p . No two pro-totypes have the same asymptotic behavior at infinity. These prototypesinclude such sequences as: 1 /n , n, n α , ln( n ) , exp( n ) , n α ln( n ) , n α / ln(ln( n )) , exp( αn . ) etc. for each nonzero real number α . Many prototypes will al-ready be familiar to those who use Big O notation [3]. The real numbers arebased on the prototypes 1 and 0.A leading term limit is a form of asymptotic limit. The definition is: If forevery positive real number (cid:15) , there is an integer N (cid:15) , a non-zero real number c , and a prototype p , such that | ( s n /p n ) − c | < (cid:15) for all integer n ≥ N (cid:15) , thenit is said that the leading term limit lim s = cp . If, instead, s n = 0 for all n ≥ N (cid:15) then lim s = 0.The second term limit is the leading term limit of s n − cp n and is written lim s .A limit is a finite sum of leading, second and higher term limits and iswritten lim s = (cid:80) i c i p .A Cauchy limit is the ordinary everyday definition of limit. If the Cauchylimit exists then it differs from the leading term limit only when the leadingterm limit is an infinitesimal.A leading limitable sequence is a sequence with a leading term limit ac-cording to the definition above. Sequences that are not leading limitablemay still have a leading term limit, but only if they can be equivalenced toa leading limitable sequence.A smooth sequence is a sequence with a limit according to the definitionabove. Sequences that are not smooth may still have a limit, but only if theycan be equivalenced to a smooth sequence. I n is used to denote an ordered field defined by a ratio of a pair of smoothsequences. It contains only infinite and infinitesimal and real numbers. ∗ R is the field of hyperreals from non-standard analysis [4] that includesboth infinite and infinitesimal numbers. Any sequence that is not smooth is called here oscillatory . A general guide-line, though by no means a precise mathematical statement, is that the limitsof smooth sequences can be written:lim x → ω f ( x ) = f ( ω )whereas limits of oscillatory sequences can’t.3are has to be taken because in the literature there are examples like [4,5]: − /
12 = ∞ (cid:88) n =1 n ; − ∞ (cid:88) n =0 n and 23 = ∞ (cid:88) n =0 ( − n The method used here does not yield counterintuitive results like these. Be-cause of ambiguities in the treatment of infinity using this notation, largelybecause it allows inappropriate use of bijection, this notation won’t be usedin this paper again.The following method for decomposing sequences into components wasinspired by Box-Jenkins time series analysis [6], but is made much easierbecause here there is a predefined map whereas in time series analysis the mapis unknown and has to be determined. In Box-Jenkins analysis a sequence ofmeasured data will typically be split into a trend, a periodic component anda random component. The periodic and random components themselves willin the most general case be non-stationary, with a varying amplitude.In the general sequence f n , start by taking n to be an integer that issufficiently large for the overall trend to be smooth, and for the oscillatorycomponent to have an amplitude that varies smoothly. (This discription isdeliberately vague, it will be firmed up after presenting examples). Onlyafter this general sequence is mapped onto a smooth sequence will the limit n → ω be equivalenced to that of the smooth sequence. The sequence couldbe written as: f n = s n + s n b n + r n where s and s are smooth mappings, b is a deterministic oscillatory mappingand r is random. I would like to use the word “ergodic” [7, 8] to describe b , but that’s too restrictive, as will be seen in the examples below. If thissequence can be decomposed as above, and if it is possible to calculate themean value of b n by the integral over the period or similar methods, and the“ensemble mean” [8] value of r n by probabilistic methods, then¯ f n = s n + s n ¯ p n + ¯ r n and the resulting mean of f will be smooth enough for the existence of aleading term limit, at least. The decomposition won’t necessarily be unique,but that doesn’t matter because the definition of limit on a smooth sequenceensures that it is closed under the operations of term by term addition andmultiplication.This introduction to oscillatory sequences suffices for the proof of animportant lemma. 4 emma I n is a proper subset of ∗ R Proof ∗ R is known to be the largest field that can be constructed fromsequences [4] and I n is an ordered field constructed from sequences [1] so I n ⊂ ∗ R . Let s n = 0 , , , , , , . . . and t n = 1 , , , , , , . . . and u n = . , . , . , . , . , . , . . . . Then under the action of ultrafilter U in ∗ R we have s = U t (cid:54) = U u but under the action of taking the mean of an oscillatory se-quence in I n , lim s = lim t = lim u so I n (cid:54) = ∗ R .Here are some examples of limits on difficult sequences. In each case theaim is to find a leading term limit, because the sequence minus the leadingterm limit yields a second sequence that can be used to find a second termlimit, etc. Examples 1 and 2 were chosen specifically as cases where Ces`arosummation [9] fails. Example b n s n = ( − e) n .Write this as b n = ( − n , s n = e n and note that ( − n is bounded andoscillatory. n has a 50% chance of being even and 50% chance of being odd sothe mean of the bounded oscillatory component is 0 . × (1) + 0 . × ( −
1) = 0and ¯ b n s n = 0 × e n = 0. Example b n = ( − (cid:98) ln( n ) (cid:99) .For sufficiently large n , the floor of ln( n ) has a 50% chance of being evenand 50% chance of being odd so ¯ b n = 0 . × (1) + 0 . × ( −
1) = 0.
Example b n = 0 , , , , , , , , , , , , , , , . . . The probability of b n having the value 1 can be calculated and turns outto be very close to 1 / ( √ n − . b n = 1 / ( √ n − .
5) and the leading termlimit is 1 / √ n . Example b n = (cid:80) mj =1 sin( d j n + e j ) where all d j are nonzero.For sufficiently large n , each of the sinusoidal terms can be integratedover its period to get a mean which in this case happens to be zero, so¯ b n = (cid:80) mj =1 Example f n is an element of the normal distribution N (0 , s ( n )).An element of N (0 , s ( n )) is the same as s ( n ) times an element of N (0 , f n = s ( n ) × Example b n is the x or y component of the H´enon strange-attractor [10] with x i +1 = y i + 1 − . x i and y i +1 = 0 . x i . 5his has a well-defined mean with ¯ x n = 0 . . . . and ¯ y n = 0 . . . . Example n to define a random real number r n ∈ U [0 ,
1) and set b n = 0 when r n is rational and b n = 1 when r n is irrational.Using Lebesgue integration [11] the mean is found to be one, so ¯ b n = 1. Example f n is the number of primes less than or equal to n .Write f n as s n ( f n / Li( n )) where s n = Li( n ) is the logarithmic integralfunction. b n = f n / Li( n ) is bounded and known to have a mean value ofone [12]. The leading term limit comes from Li( n ) ∼ ln( n ) /n . Example f n = tan( n ) = b n s n . This is tough because not only is tan unbounded, itsstandard deviation is also unbounded and the expected value of the standarddeviation actually increases with n . Set s n = √ n and b n = tan( n ) / √ n . Then b n is oscillatory with a bounded oscillatory standard deviation that does nottend to zero. By symmetry of tan( n ) about zero the mean ¯ b n = 0. Example
10. Two closely related examples with the same leading termlimit.10a. f n = tan( n (mod π/ )10b. x ∈ U [0 , π/
2) , f n = tan( x )This was sufficiently difficult that it was worth checking three ways, bydirect and Monte-Carlo simulation, by integration over a period, and by useof order statistics [13]. Despite the function, median, quartiles etc. beingtime independent, the mean is time dependent because with an increasingnumber of values of n the singularity at π/ Figure 1: Direct simulation of 10a. Upper and lower values of the mean.Order statistics of 10b yields n (cid:80) nk =1 tan( kπ/ n + 1)).6ntegration over a period for both 10a and 10b yields (cid:82) − α/n tan(2 x/π ) dx where α ≈ lim f ( i ) = (2 /π ) ln( n ).From the above ten examples it can be seen that the limit of the oscilla-tory component b n as defined above at sufficiently large but finite number n exists for a remarkably large range of types of oscillatory sequences. In everyexample above ¯ f n is a smooth sequence. Conjecture : Every sequence has a well-defined leading limit.The above examples provide enough information to start attacking thisconjecture. The following is a generalized methodology for generating sucha limit. • Recall that f n = f ( n ) where f is a mapping. Periodics • Start with simply periodic functions where there exists a real number x such that f ( x + x ) = f ( x ) for all necessary x . • Fold the integers at period x to get set F ⊂ [0 , x ) and carry out aLebesgue integration [11] of f ( x ) over F . • If the Lebesgue integral exists then the mean ¯ f is the Lebesgue integraldivided by the Lebesgue measure of F . • If the Lebesgue integral does not exist because of singularities in f ( x )at a finite number of points in F ⊂ [0 , x ) then use order statistics [13]to calculate how close x gets to those singularities as a function of n and use that expected distance as a bound on the integral. The integralwill then have a limit on I n as a function of n and so the mean ¯ f ( n )defined as the Lebesgue integral over the Lebesgue measure of F willbe well-defined. • The next class of sequences comes from transformed periodic mappings f ( g ( x )) where f is periodic and g is leading term limitable. The meanof f ( g ( x )) is taken to be equal to the mean of f ( x ). • The next class of sequences h ( n ) is of those that tend to a simplyperiodic or transformed periodic mapping. If f ( g ( n )) has a mean thatis leading term limitable using prototype p then this is equivalent tosaying that the Cauchy limit lim n →∞ (( h ( n ) − f ( g ( n )) /p n ) = 0. The7ean of h ( n ) is then taken be the mean of the simply periodic ortransformed periodic sequence that it tends to. The phrase “periodicsequence” below means h ( n ). Randoms • Let f ( i ) be random. The mean and other statistical parameters mayvary arbitrarily with i . From a sufficiently large number of actualisa-tions take the ensemble mean [8] at given i = n . • If the ensemble mean is not easy to find (which sometimes happenswhen f ( i ) is unbounded) then use order statistics to estimate how large f ( i ) is likely to be at given i . This will likely allow an integral to befound for the ensemble mean. • Treat transformed randoms and sequences that tend to randoms in thesame way as for periodics.
Chaos • Chaotic sequences will tend to a strange attractor. The best-knownstrange attractors have a well-defined mean. • Treat variations on this in exactly the same way as variations on peri-odic functions.
Sums • The method used to calculate the ensemble mean of a random sequenceat given n will probably generate a smooth sequence. The productof two smooth sequences is a smooth sequence. For full generalityof oscillatory sequences each periodic sequence and chaotic sequenceneeds to be multiplied by a smooth sequence to recover smooth trendsin amplitude. The limit of the product of smooth sequences is theproduct of the limits. • The ensemble mean of a random sequence as a function of n is notnecessarily smooth. If a non-smooth ensemble mean can be convertedto a finite sum of smooth times periodic and smooth times chaotic se-quences (term by term multiplication and addition) then a well-definedlimit exists. If not then some other method would be needed to findthe limit, such as the imposition of symmetry conditions. By defini-tion, the ensemble mean of an ensemble mean is identical to the originalensemble mean. 8 So to summarise, it is very likely that a general sequence has a well-defined limit if it can be decomposed into a finite sum of: smoothsequence, smooth sequence times periodic sequence, smooth sequencetimes chaotic sequence, and random sequence – providing the ensemblemean of the random sequence can be decomposed into a finite sumof: smooth sequence, smooth sequence times periodic sequence, andsmooth sequence times chaotic sequence.
Lemma
The complete set of sequences that can be mapped by this methodto a smooth sequence are closed under addition and multiplication.
Proof
Under addition because the sum of two finite sums is a finite sum.Under multiplication because, temporarily leaving aside the issue of inverses:the product of two smooth sequences is smooth, the product of two periodicsequences is periodic, the product of two chaotic sequences is chaotic, theproduct of two random sequences is random, the product of a periodic andchaotic sequence is chaotic, the product of a smooth and random sequenceis random with smooth ensemble mean, the product of a periodic and ran-dom sequence is random with periodic ensemble mean, and the product ofa chaotic and random sequence is random with chaotic ensemble mean, andthe cases of smooth times periodic and smooth times chaotic are explicitlytaken care of in the definition. When inverses are present, products becomesimplified.In mathematical notation:a) Split the sequence f into smooth s , periodic p , chaotic k and random r components: f ( n ) = s ( n ) + (cid:88) j j s ( n ) j p ( n ) + (cid:88) k k s ( n ) k k ( n ) + (cid:88) l l r ( n )where each sum is finite and each appearance of s can mean a differentsmooth sequence.b) Take the ensemble mean of each random component and split it intosmooth, periodic and chaotic components: l ¯ r ( n ) = l s ( n ) + (cid:88) m m s ( n ) m p ( n ) + (cid:88) i i s ( n ) i k ( n )where again each sum is finite and each appearance of a sequence can be adifferent sequence. 9) Take the mean of each periodic and chaotic component. The mean willbe dependent on n if and only if order statistics are required. Let’s write itas if it is independent of n . l ¯¯ r ( n ) = l s ( n ) + (cid:80) m m s ( n ) m ¯ p + (cid:80) i i s ( n ) i ¯ k ¯ f ( n ) = s ( n ) + (cid:80) j j s ( n ) j ¯ p + (cid:80) k k s ( n ) k ¯ k + (cid:80) l l ¯¯ r ( n )d) Then ¯ f ( n ) will be a finite sum of smooth functions which is usuallysmooth, and we can take the limit using the techniques described earlier:lim n → ω f ( n ) = lim n → ω ¯ f ( n )This gives the mapping from a general sequence to a smooth sequence whichallows euivalencing of the limits of the two.Note that periodic and chaotic components are treated in exactly thesame way, except that the mean of a strange attractor is not calculated fromthe integral over a period. For practical use, the leading term limits of eachcomponent can be used to get the leading term limit of f and then thatsubtracted from f gives a new sequence that can be used to give the secondterm limit, etc. What about the limit of a function at a real number?When a function f is continuous at x = a ∈ R then there’s no problem.lim x → a f ( x ) = f ( a ).When f is not continuous at x = a then the following method works:Allow (cid:15) to tend to a specific infinitesimal (cid:15) . Split f ( x ) near a into f ( x ) = f ( x ) + f ( x ) f ( x ) where f and f are monotone continuous near a + (cid:15) and/or a − (cid:15) and f is ergodic [7] at the same point(s). Take the meanvalue of the ergodic component to get ¯ f . If f ( x ) has a value on the realnumber line at both a + (cid:15) and a − (cid:15) then use:lim x → a f ( x ) = lim (cid:15) → (cid:15) (1 / f ( a + (cid:15) ) + f ( a + (cid:15) ) ¯ f ( a + (cid:15) ) + f ( a − (cid:15) ) + f (2( a − (cid:15) ) ¯ f ( a − (cid:15) ))Otherwise, if f doesn’t exist on the real numbers at either a + (cid:15) or a − (cid:15) ,use the one-sided limit:lim x → a f ( x ) = lim (cid:15) → (cid:15) ( f ( a ± (cid:15) ) + f ( a ± (cid:15) ) ¯ f ( a ± (cid:15) )10his leads to results such as the following.lim x → π/ tan( x ) = 0 lim x → (cid:15) ln( x ) = − ln(1 /(cid:15) ) lim x → /ω | / sin( x ) | = ω lim x → cos(1 /x ) /x = 0What about the complex numbers? There’s no trouble with writing a com-plex number as z = a + ib where a and b are infinite or infinitesimal numbers,but this doesn’t immediately help much in determining the infinite limits ofuseful complex functions.Let’s look at the infinite limits of e z . This is smooth and the limits are e ω and e − ω at the ends of the real number line. e z is periodic with mean zero inthe imaginary direction so has limit 0 at both ends of the imaginary numberline. Further, e z is periodic with mean zero along every straight line passingthrough the origin other than the real number line.To get the transition from e ω to 0 consider the complex number z = ω + ib for real b. e ω + ib = e ω (cos( b ) + i sin( b )).The mean of a function is the integral (along a contour at constant dis-tance from z = 0) divided by the interval. So the limit of e ω + ib is e ω (sin( b ) + i (1 − cos( b ))) /b .What about the limit of a complex function at a finite complex number?On the complex numbers, calculate the mean from the average around acontour at infinitesimal radius (cid:15) .lim z → z f ( z ) = lim (cid:15) → (cid:15) (cid:72) f ( z ) ds/ (cid:72) ds .When s is the length of a circular path of radius (cid:15) surrounding z : s = (cid:15)θ . (cid:72) f ( z ) ds = (cid:82) θ max f ( z + (cid:15)e iθ ) (cid:15)dθ . (cid:72) ds = θ max (cid:15) .lim z → z f ( z ) = lim (cid:15) → (cid:15) θ max (cid:82) θ max f ( z + (cid:15)e iθ ) dθ To see how it works, let I = (cid:82) θ max f ( z ) dθ and consider f ( z ) = z α .When z (cid:54) = 0, f ( z ) = ( z + (cid:15)e iθ ) α = z α + αz α − (cid:15)e iθ + O ( (cid:15) ) I = z α θ max + (cid:15) (cid:82) θ max αz α − e iθ dθ + O ( (cid:15) )lim z → z f ( z ) = lim (cid:15) → (cid:15) I/θ max so the leading term limit when z (cid:54) = 0 is lim z → z z α = z α When z = 0 and α (cid:54) = 0, 11 ( z ) = (cid:15) α e iαθ I = (cid:15) α iα (cid:2) e iαθ (cid:3) θ max As with reals, take the mean of an oscillating function over a period. θ max = 2 π/α lim z → z α = 0When α = 0, lim z → z s as path length differs from normal contour integration. It is im-portant not to get the two mixed up. (cid:72) f ( z ) ds = (cid:82) θ max f ( z + (cid:15)e iθ ) (cid:15)dθ . (cid:72) f ( z ) dz = (cid:82) θ max f ( z + (cid:15)e iθ )( i(cid:15)e iθ ) dθ . Evaluating definite integrals using limits of series requires care and wheneverfeasible the problem that yields the integral should be examined for clues tothe behaviour near infinity and near each singularity.When no such information is available and when an indefinite integral isavailable, substitute ω for infinity and when the limit is at a finite singularity x substitute x ± /ω .A Riemann sum can often be used to calculate the leading term limit ofa definite integral. Terms beyond the leading term limit should sometimesbe treated with suspicion and where high accuracy is needed a cubic splinefit [14] would be better. The following Riemann sum can be used for anyfunction without singularities on domain [0 , ω ). (cid:82) ω f ( x ) dx = lim m → ω (cid:0) m lim n → mω (cid:80) nk =1 f (( k − . /m ) (cid:1) = lim n → ω √ n (cid:80) nk =1 f ( x ) when x = ( k − . / √ n For example, using the new definition of “lim”, this immediately yields theexact results: (cid:82) ω dx = ω (cid:82) ω xdx = ω / (cid:82) ω cos( x ) dx = 0 (cid:82) ω sin( x ) − x cos( x ) dx = 0More commonly it yields approximate results: (cid:82) ω exp( x ) dx = exp(1 / ω )(exp( ω ) − ω (exp(1 /ω ) − ≈ (1+1 / ω +1 / ω )(exp( ω ) − / ω +1 / ω ) ≈ exp( ω )12his has the correct leading term limit, with an error term smaller than theleading term by a factor of O ( ω ). That’s a typical order of magnitude factorfor an error term of a Riemann sum.When there is a singularity on the reals, a transformation can help. Let x = t ( ξ ) where ξ is divided into equal intervals and each interval is evaluatedat its midpoint. The width of each x interval is ∆ x = t (cid:48) ( ξ )∆ ξ . Then (cid:90) x max x min f ( x ) dx = lim n → ω √ n k max (cid:88) k = k min f ( t ( ξ )) t (cid:48) ( ξ )where ξ = ( k + k min − . / √ n , and x min = t ( k min ) and x max = t ( k max ).Change the sign if limits are swapped.Useful transformations include: • t ( ξ ) = x min + 1 /ξ . This maps (0 , ω ) to ( x min , ω ) for functions thatmisbehave at x min but integrate to a finite number at the infinite limit. • t ( ξ ) = x min + ξ/ (1 − exp( − ξ )) This maps ( − ω, ω ) to ( x min , ω ) for func-tions that misbehave at x min and infinity. • t ( ξ ) = ( x max + x min ) / x max − x min ) tan − ( πξ/ ( x max − x min )) /π . Thismaps ( − ω, ω ) to ( x min , x max ) for functions that misbehave at both lim-its. I don’t want to deal in any detail with generalized functions in this paper,but there’s no difficulty in squaring the Dirac delta function [15]. For gen-eralized functions that can be constructed from limits, everything remainscommutative and there’s no need for non-commutative algebras.Let f ( x ) be any probability density function on the real numbers. Let (cid:15) be any appropriate infinitesimal number. Then the Dirac delta function isthe set of functions defined by δ ( x ) = lim (cid:15) → (cid:15) f ( x/(cid:15) ) /(cid:15) . To see that thelimit exists, express it as the limit of a smooth sequence as follows. Let g ( ω ) = − /(cid:15) , then the limit becomes lim n → ω g ( n ) f ( xg ( n )). Call this limit f ( x/(cid:15) ) /(cid:15) . The square of the Dirac delta function is simply f ( x/(cid:15) ) /(cid:15) . Toget the correct value of the Heaviside function at zero, H (0) = 0 .
5, the initialprobability density function should be symmetric about zero.13
Conclusions
I hope that the methods of this paper go some way towards banishing theconcept of divergence. Methods are presented here for evaluating the limitof sequences that converge/diverge to infinity, for evaluating limits of oscil-latory and general sequences, for finding limits on the reals, finding limits oncomplex numbers, and for evaluation of improper integrals.Without exception, each of the methods used here is “local” on the in-tegers and real numbers in the sense that the mean is defined at a specificinteger or real number without recourse to adjacent integers or real numbers.The ensemble mean of a sequence at given index n is totally independent ofthe values of the sequence at values n − n + 1. This takes some gettingused to, and is possible only by using the properties of the mapping that isused to generate the sequence. On complex numbers this restriction is re-laxed to defining the mean using an integral over a circle of fixed infinitesimal“local” radius.This part of this paper leaves several questions unanswered. Althoughproofs aren’t needed for practical application of the methods presented here,ideally proper proofs/disproofs should be found for the conjectures: • All infinite sequences can be decomposed into a finite sum of smooth,periodic, chaotic and random components and this decomposition yieldsa unique leading term limit. • Every definite integral on the reals has a unique value on I n .As a piece of applied mathematics, the number one priority would be incodifying the methods presented here and extensions of them into software,so that someone could ask, for example “ (cid:82) ∞ sin( x ) dx = ?”, and get back animmediate exact answer. References ∼ ∼ ∼∼