Quence from the partial sums. The SM is associated for the
Quence of the partial sums. The SM is connected towards the generalized definition T of series and arises in the study of polynomial approximations [16]. An instance on the SM considers the Taylor expansion of the geometric series 0 x k . k= Within this context, the -sum of the sequence ( x k )kN for x R is 1/(1 – x ), if – x 1, exactly where three.5911. For other values of x R, the sequence ( x k )kN isn’t -summable. This instance shows that the SM is able to assign a worth to a larger number of series than the SM by Abel along with the N lund means, due to the fact 1. 4.3.four. Oscillatory Very simple Finite Sums Alabdulmohsin [16] derived a technique analogous towards the EMSF for dealing with oscillating sums. Within the following, all series should be interpreted inside the context on the generalized definition T of series. Offered an alternating series 0 (-1) g(), where for each and every point x0 [0, ), the = function g is analytic on some open disc centered at x0 , along with the beginning point is offered by the following formal expression, equivalent to the Moveltipril custom synthesis T-value limit L for infinite sums: L=r =Nr (r) g (0), r!where Nr ==(-1) r ,(154)where the IEM-1460 web convention 00 = 1 is required. The constants Nr , which could be thought of as an alternating analog of the Bernoulli numbers (as well as with the Euler numbers), can be obtained by way of the recurrence equation N0 = 1 , 2 and Nr = – 1 r -1 r N , if r 0 . 2 r =0 (155)-1 For an alternating SFS offered by f (n) = F r n=0 (-1) g(), T-summable to a value L C, the exceptional generalization f G (n), which agrees with the polynomial approximation process presented in (140), is offered, formally, byf G (n) =r =Nr Nr (r) g (0) + (-1)n+1 g(r) (n) . r! r! r =(156)The Equation (156) is an analog from the EMSF for the case of alternating sums. Also, for all n C, it’s valid that f G (n) ==(-1) g() – F r (-1) g() .=n(157)Mathematics 2021, 9,30 ofExamples from the use on the Equation (156) would be the following closed formulae for alternating power sums: n -1 1 (-1)n Fr , (158) (-1) = – 2 two =Frn -1 =(-1) = – four + (-1)n+r =2n + 1 , 4 r N nr- ,(159)Frn -1 =(-1) r = Nr + (-1)n+(160)exactly where Equation (160) offers a periodic analog of Faulhauber’s formula (59). As a consequence of Equation (156), if a offered alternating series 0 (-1) g() is = T-summable to some worth L, where g : C C is usually a function of a finite polynomial order m, then the value L is often obtained by L = limFrn -1 =n(-1) g() + (-1)nFrr =mNr (r) g (n) . r!(161)In other words, the alternating SFS following asymptotic expression:Frn -1 =-1 n=0 (-1) g() may be represented by the(-1) g() L + (-1)n+r =mNr (r) g (n) , r!(162)where the last term tends to 0 when n . This supplies a approach to derive asymptotic expressions for alternating series, from which it can be achievable to extract an sufficient worth for any provided divergent series and, in some instances, to derive analytic expressions for divergent alternating series. As an example, applying the generalized definition T for the alternating series 0 (-1) log(1 + ), it can be probable to acquire L = log(2/ ) /2. = 4.three.five. Oscillatory Composite Finite Sums The analogue in the EMSF offered in Equation (156) for alternating SFS is usually general-1 ized to alternating CFS (OCFS with period p = 1) of the kind f (n) = F r n=0 (-1) g(, n), as follows: The exclusive all-natural generalization f G (n) of f (n), which agrees using the polynomial approximation approach (140), is offered by: f G (n) =r =Nr r!r g(t, n) trt =+ (-1)n+r =Nr r!r g(t, n) trt=n.(163)Equation (163) allows acquiring closed expressions as well as a.
