New Expansion and Infinite Series

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1 Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, HIKARI Ltd, New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University of Posts nd Telecommunictions Nnjing, P.R. Chin Copyright c 204 Diyun Zhng. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. Abstrct Different from the Tylor polynomils, new formul for function expnsion is proposed where the terms re not polynomils. A new infinite series bsed on the new formul is lso proposed, nd the new infinite series cn keep some importnt properties of the originl functions. Some forms of reminder re lso presented for nlysis of convergence. In order to show some internl reltionships between the new results nd Tylor s theorem, some importnt theorems hve been proved in this pper. Finlly, Some exmples re given nd the regions of convergence for the new infinite series re nlyzed, the results show tht the region of convergence is much lrger thn tht obtined by Tylor s Series. Mthemtics Subject Clssifiction: 4A58, 4A20, 4A30, 40A30 Keywords: Expnsion of function, Tylor expnsion, Tylor series, Reminder nlysis, Convergence, Region of convergence Introduction It is well known tht Tylor s theorem [, 2] hs mny pplictions in mthemtics, physics, engineering nd other fields. Up to now, mny reserches re lso relted with Tylor s expnsion or Tylor s series. For exmple, the pproximte method for solving nonliner differentil equtions [3]. In [4], severl exmples were presented for ordinry differentil eqution pplictions
2 062 Diyun Zhng tht cn be solved successfully using the Tylor series method. In [5], A numericl method for solving prtil differentil equtions bsed upon the Tylor series ws proposed. Literture [6] presented Tylorseries expnsion method for second kind Fredholm integrl equtions system. In [7], by using specil property in the integrl representtion of the reminder vlue of the Tylor series expnsion, n expnsion for nlytic functions ws introduced. In literture [8], the reserch explicitly obtin the reltion between the coefficients of the Tylor series nd Jcobi polynomil expnsions. Although some pplictions hve been mentioned bove, there re mny other pplictions bout Tylor s expnsion or series. Let f(x) be given function, The results mentioned bove re ll relted with originl form of Tylor s expnsion or series, in which the coefficients of ech term re lwys the constnt derivtives f (k) () t point, nd no mtter how mny terms of Tylor s expnsion you choose, they re kind of polynomils of x when neglecting the reminder. Although the Tylor theorem is importnt nd hs mny pplictions, we hve to sk question: re there ny other forms of expnsion for given function f(x) besides the originl form of Tylor s expnsion? The nswer is yes. In this pper, new form of expnsion is proposed, which is different from the originl form of Tylor s expnsion. In generl, the new expnsion generted by given function f(x) is not the form of the Tylor polynomils of the vrible x. For the given function f(x), the terms generted by the new expnsion re of functions ssocited with the derivtives f (k) (x) of the first k orders, insted of the constnt derivtives f (k) () t point. In some cses, the expnsions generted by the theorem proposed in this pper re rtionl functions, insted of polynomil functions. In mny pplictions, we hve to del with the functions with poles, such s rtionl functions, logrithmic functions nd so on, but the polynomil functions cn not express the properties of poles nd corresponding singulr prts properly. A new infinite series bsed on the expnsion is lso proposed in this pper, nd the region of convergence of the infinite series is nlyzed. Some ppliction exmples showed tht the region of convergence is much lrger thn tht obtined by the originl form of Tylor s series directly. Although there re mny wys to pproximte given function f(x) by polynomils, rtionl functions or some other forms of functions, the method proposed in this pper is specil one, depending on the functions of derivtives. The new expnsion nd infinite series my be widely used in the fields to which the originl form of Tylor s expnsion nd infinite series cn be pplied. Using the new results, we my obtin more chievements in mny fields.
3 New expnsion nd infinite series New Formul of Expnsion Let f (x) be function, the entire rel line is denoted by R. We define C n [A, B] to be the set of functions for which f (n) (x) exists nd is continuous on the closed intervl [A, B] R. An importnt theorem concerning new expnsion for function is proposed in this pper nd given in the following: Theorem 2.. Let f (x) be function hving finite (n + )th derivtive f (n+) (x) everywhere in n open intervl (A, B) nd ssume tht f (x) C n [A, B]. Assume tht [A, B]. Then, for every x in [A, B], we hve f (x) = Z n [f (x; )] + R n [f (x; )] () where Z n [f (x; )] = f () + R n [f (x; )] = ( ) k f (k) (x) (x ) k (2) ( t) n f (n+) (t) dt (3) where Z n [f (x; )] is known s the expnsion of function f (x) up to the nth derivtive t the point x =, nd R n [f (x; )] is known s the reminder of the expnsion Z n [f (x; )] t the point x =. Proof. Let us consider the following integrl: I n = integrting by prts we find tht I n = (t ) n f (n+) (t) dt = (t ) n f (n+) (t) dt (4) (t ) n df (n) (t) = (x )n f (n) (x) x (t ) n f (n) (t) dt (n )! From (5) we find the recursive formul in the following: (5) I n = (x )n f (n) (x) I n (6) From (4) we hve I 0 = f (x) f () (7)
4 064 Diyun Zhng From (6) we hve I n = (x )n f (n) (x) When n is odd, using (8), we hve (n )! (x )n f (n ) (x) + I n 2 (8) I n = f (n) (x) (x ) n f (n ) (x) (n )! (x )n + + f (x) (x ) I 0 (9) Hence I 0 = ( ) k f (k) (x) (x ) k I n (0) Therefore, when n is odd, from (4), (7) nd (0) we hve f (x) = f () + When n is even, from (8) nd (6), we get ( ) k f (k) (x) (x ) k + ( t) n f (n+) (t) dt () I n = f (n) (x) (x ) n f (n ) (x) (n )! (x )n + f (x) (x ) + I 0 (2) then I 0 = ( ) k f (k) (x) (x ) k + I n (3) Therefore, when n is even, from (4), (7) nd (3) we hve f (x) = f () + ( ) k f (k) (x) (x ) k + ( t) n f (n+) (t) dt (4) From (4) nd (), we cn conclude tht, no mtter wht the number of n my be (odd or even), the following lwys holds: f (x) = f () + ( ) k f (k) (x) (x ) k + ( t) n f (n+) (t) dt (5) This completes the proof. If ignoring the reminder of (5), we hve the following pproximte formul f (x) Z n [f (x; )] = f () + ( ) k f (k) (x) (x ) k (6)
5 New expnsion nd infinite series 065 Formul (2) (or(6)) is different from Tylor s theorem. The righthnd side of formul (6) is new form of expnsion generted by given function f(x) t point. Obviously, f (k) (x) is usully not polynomil for given function f(x), the righthnd side of formul (6) is usully not polynomil either, therefore, we obtin new form of expnsion generted by given function f(x) t point. Tylor s theorem gives n pproximtion of k times differentible function round given point by kth order Tylor polynomil, which pproximtely determines the function in some neighborhood of the point. But our formul (see(2) or (6)) shows tht, in some pplictions, we give n pproximtion of k times differentible function round given point by kind of function rther thn polynomil, for exmple, some rtionl functions re obtined (see exmples of section 6). Obviously, rtionl function is usully different from polynomil, therefore, our formul (2) (or(6)) is new formul. It should be noted tht, in our formul (2) or (6), the coefficients ( ) k f (k) (x) / (k =, 2,, n) of terms (x ) k re functions of vrible x, which is different from the constnt coefficients f (k) ()/ obtined by Tylor s expnsion. 3 Forms of Reminder 3. The First Form of Reminder When x >, ( x) n never chnges sign on [, x] nd ssume f (n+) (x) is continuous on [, x], then, using the generlized MenVlue Theorem [] in (3), we hve R n [f (x; )] = ( t) n f (n+) (t) dt = ( ) n f (n+) (ξ) (n + )! (x )n+ (7) Or R n [f (x; )] = f (n+) (ξ) x n+ (8) (n + )! where ξ lies between nd x. When x <, ( x) n lso never chnges sign on [x, ], we lso hve the sme results like (7) nd (8). When x =, the righthnd side of integrl (3) is 0, the righthnd side of (8) is lso 0. Therefore, no mtter wht hppens, (8) lwys holds.
6 066 Diyun Zhng 3.2 The Second Form of Reminder Since ( t) n is continuous function of t, ssume f (n+) (t) is continuous on [, x], using the MenVlue Theorem in (3), we hve R n [f (x; )] = ( t) n f (n+) (t) dt = ( ξ)n f (n+) (ξ) (x ) (9) where ξ lies between nd x. 3.3 The Third Form of Reminder Since ( t) n is continuous function of t, ssume f (n+) (t) is continuous nd never chnges sign on [, x], by using the generlized MenVlue Theorem in (3), we hve R n [f (x; )] = ( t) n f (n+) ( ξ)n (t) dt = f (n+) (t) dt (20) ( ξ)n ( = f (n) (x) f (n) () ) where ξ lies between nd x. 4 Infinite Series Bsed on the New Expnsion Suppose tht f (x) is function ll of whose differentil functions f (k) (x) re continuous in n intervl surrounding the point x =. Let n in formul (2), we cn certinly form the infinite series in the following nd denoted by Z [f (x; )]: Z [f (x; )] = f () + ( ) k f (k) (x) (x ) k (2) The expression (2) is clled the infinite series bsed on the new expnsion (2), which is generted by f (x) t. If = 0, we hve Z [f (x; 0)] = f (0) + ( ) k f (k) (x) x k (22)
7 New expnsion nd infinite series Convergence Anlysis of the New Infinite Series We hve proved tht the formul of reminder (error term) could be expressed s integrl (3), in which f (n+) (x) is continuous. Therefore, we hve the following theorem. Theorem 5.. If f (x) is infinitely differentible in set S, x S, S, the series (2) converges to f (x) if nd only if integrl (3) tends to 0 s n, i.e. lim R n [f (x; )] = 0 (23) Integrl (3) is clled the integrl form of the reminder. lim R n [f (x; )] is lso denote by R [f (x; )]. The form of integrl (3) enbles us to give the following sufficient condition for convergence of the series proposed in this pper. Theorem 5.2. Assume f (x) is infinitely differentible in set S, x S, S, nd ssume tht there is positive constnt M such tht f (n) (x) M, n =, 2,, x S, S (24) then the series generted by f (x) t (see(2)) converges to f (x) for ech x S, S, i.e. f (x) = Z [f (x; )], x S, S (25) Proof. Using (8) nd (24), we obtin the estimte R n [f (x; )] M (n + )! x n+ (26) But x n+/ (n + )! 0 s n for ech finite x S, S, hence R [f (x; )] = lim R n [f (x; )] = 0 (27) i.e. R [f (x; )] = 0, x S, S (28) therefore, f (x) = Z [f (x; )] + R [f (x; )] = Z [f (x; )], x S, S. In order to show some internl reltionships between the new results proposed in this pper nd Tylor s theorem, we write Tylor s theorem in the following: f (x) = T n [f (x; )] + E n [f (x; )] (29)
8 068 Diyun Zhng where T n [f (x; )] = f () + f (k) () (x ) k (30) E n [f (x; )] = (x t) n f (n+) (t) dt (3) We denote lim T n [f (x; )] nd lim E n [f (x; )] by T [f (x; )] nd E [f (x; )] respectively. Tylor s series cn be written in the following: T [f (x; )] = f () + f (k) () (x ) k (32) Theorem 5.3. When x S, S, if R [f (x; )] = 0 nd E [f (x; )] = 0, then the following holds f (x) = lim Z n [f (x; )] = lim T n [f (x; )] (33) Or f (x) = Z [f (x; )] = T [f (x; )] (34) Proof. Since R [f (x; )] = 0, x S, S, from Theorem 5. we hve f (x) = Z [f (x; )], x S, S. Let n, from (29) we hve f (x) = T [f (x; )] + E [f (x; )] = T [f (x; )] Therefore, expressions (33) nd (34) hold. Theorem 5.3 indictes tht, if the conditions of Theorem 5.3 re stisfied, the series Z [f (x; )] (proposed in this pper) nd Tylor s series T [f (x; )] re equl nd they both converge to f (x) for ech x S, S. Generlly speking, if x / S, the series Z [f (x; )] (proposed in this pper) will hve different vlues from Tylor s series T [f (x; )], which indicte tht the series (proposed in this pper) is new kind of infinite series. More detil will be given in Exmple 6. nd Exmple 6.2. Theorem 5.4. Let M be positive constnt, 0 < M <, if f (n) (x) < M for ny x (, ), n =, 2,, then the following holds: f (x) = Z [f (x; )] = T [f (x; )], (, ), x (, ) (35) Proof. Let S = (, ), from Theorem 5.2 we hve f (x) = Z [f (x; )], (, ), x (, ). On the other hnd, let ξ = +θ (x ), 0 < θ <,
9 New expnsion nd infinite series 069 then ξ must lie between x nd, using the MenVlue Theorem in (3), we hve E n [f (x; )] = = (x t) n f (n+) (t) dt = (x ξ)n f (n+) (ξ) (x ) (x )n+ ( θ) n f (n+) ( + θ (x )) (36) Since 0 < θ <, f (n+) ( + θ (x )) < M nd (x ) n+ / 0 s n, then we obtin E [f (x; )] = 0. Let n, from (29) we hve f (x) = T [f (x; )]+E [f (x; )] = T [f (x; )], (, ), x (, ) Therefore, (35) holds. 6 Exmples Some exmples will be given in the following for describing the pplictions of the results proposed in this pper. Exmple 6. Let x, > 0, x > 0, f (x) = ln (x), using Theorem 2., we hve: ln (x) = ln () + ( ) k ln (k) (x) (x ) k + R n [f (x; )] (37) where R n [f (x; )] = ( t) n ln (n+) (t) dt (38) Since ln (k) (x) = ( ) k (k )!x k (39) Then, formul (37) cn be written in the following ln (x) = ln () + ( ) k x + k x The infinite series Z [f (x; )] is Z [f (x; )] = Z [ln (x; )] = ln () + (t ) n t (n+) dt (40) ( ) k x (4) k x
10 070 Diyun Zhng Using (9), we hve R [f (x; )] = lim R n [f (x; )] = lim R n [ln (x; )] = lim ( t) n ln (n+) (t) dt = lim (( ) n ( ξ) n ) (x ) ξ (( n+ ) n ) θ (x ) (x ) = lim + θ (x ) ( + θ (x )) (42) where 0 < θ <. Since > 0, 0 < θ <, when θ (x ) > /2, the vlue of (42) tends to 0, or when x > ( 2θ = ) (43) 2θ is stisfied, we get R [f (x; )] = lim R n [f (x; )] = 0. Since > 0, 0 < θ <, the rnge of ( /(2θ)) (see the righthnd side of (43)) is (, /2), if we choose ( x ) = (44) 2 2 then (43) must be stisfied. Therefore, when x /2, > 0, the function ln (x) cn be expressed by the following convergent series. ln (x) = Z [ln (x; )] = ln () + Let =, we hve ( ) k x, x /2, > 0 (45) k x ln (x) = ( ) k x, x [/2, ) (46) k x The Tylor series for function ln (x) cn be written in the following ln (x) = (x ) (x )2 2 + = ( ) k+ (x ) k, x (0, 2] (47) k The region of convergence of Tylor s series is x (0, 2], which is much less thn the region (x [/2, ) ) obtined in this pper.
11 New expnsion nd infinite series 07 If x ([/2, ) (0, 2]), or x [/2, 2], the series (46) nd Tylor s series (47) re equl nd converge to ln (x) (see Theorem 5.3), but when x / ([/2, ) (0, 2]) or x / [/2, 2], the series (46) nd Tylor s series (47) will hve different vlues, therefore, they re different functions. Generlly speking, the series Z [f (x; )] (proposed in this pper) is different from Tylor s series T [f (x; )], which indicte tht the series Z [f (x; )] (proposed in this pper) is new kind of infinite series. On the other hnd, lthough the point x = 0 is divergent both in the righthnd side of series (46) nd (47), series (46) shows n importnt property tht the point x = 0 is pole of the series Z [ln (x; )] (see the righthnd side of (46)), which is lso the pole of ln (x), while in series (47), the pole of ln (x) is not the pole of Tylor s series (see the righthnd side of (47)). It implies tht Tylor s series my lose some importnt properties of the originl functions. Exmple 6.2 Let f (x) = /(x + b), b, from (), (2), (3) we hve where From (9) we hve ( R n [f (x; )] = Then R [f (x; )] = lim f (x) = Z n [f (x; )] + R n [f (x; )] (48) Z n [f (x; )] = + b (( θ (x ) + θ (x ) + b (x ) k (x + b) k+ (49) ) n (n + ) (x ) ( + θ (x ) + b) 2 (50) θ (x ) ) n (n + ) (x ) ) + θ (x ) + b ( + θ (x ) + b) 2 where 0 < θ <. If +b > 0, when θ (x ) > /(2 ( + b)), from (5), we hve lim R n [f (x; )] = 0, or when x > (52) 2 ( + b) θ is stisfied, the following series (53) is convergent. Z [f (x; )] = lim Z n [f (x; )] = + b (x ) k (5) (x + b) k+ (53) Since 0 < θ <, the rnge on the righthnd side of (52) is (, /(2 ( + b))), if we choose x (54) 2 ( + b)
12 072 Diyun Zhng then the inequlity (52) lwys holds. Therefore, when + b > 0, the series (53) generted by function f (x) = /(x + b) is convergent if (54) holds. If + b < 0, when θ (x ) < /(2 ( + b)), we hve lim R n [f (x; )] = 0 from (5), or when x < (55) 2 ( + b) θ is stisfied, the series (53) is convergent. Since 0 < θ <, the rnge on the righthnd side of (55) is ( /(2 ( + b)), ), if we choose x (56) 2 ( + b) then the inequlity (55) lwys holds. Therefore, when + b < 0, the series (53) is convergent if (56) holds. For exmple, let = 0 nd b =, then + b > 0, from (54) we hve x /2, nd (53) becomes the following convergent series with rtionl terms: + x = x k, x /2 (57) k+ ( + x) which is different from Tylor s series in the following: + x = + ( ) n x k, < x < (58) It is well known tht the region of convergence of Tylor s series for the function /( + x) is x (, ), which is much less thn the region x [ /2, ) obtined by series (57). Similr to the result of Exmple 6., when x / ([ /2, ) (, )) or x / [ /2, ), series (57) nd (58) re different functions. On the other hnd, the righthnd side of (57) still hs the sme pole s the function /( + x), but the righthnd side of (58) loses the importnt property. 7 Conclusions The focus of this pper is on the generl principle of the new form of expnsion nd infinite series for given function, which is different from the Tylor polynomils nd infinite series generted by the sme given function. As new theory for expnsion nd infinite series, the work ccomplished in this pper is fvorble for the frther corresponding reserches. Using the results proposed in this pper, we my obtin more chievements in mny fields. Besides, the reserch methods nd the results presented in this pper re
13 New expnsion nd infinite series 073 cdemiclly vluble nd hve significnces in mny theories nd pplictions, not only in mthemtics but lso in science nd engineering. It is well known tht Tylors theorem hs mny pplictions in mny fields. It implies tht the new expnsion nd infinite series proposed in this pper cn be extended to lmost ny fields to which the originl form of Tylors expnsion nd infinite series cn be pplied. For exmple, we hope tht this new theory cn be extended to the fields of complex nlysis, pproximtion, numeric nlysis, engineering, physics, informtion science nd so on, which wit for deeply investigting nd exploring. References [] G. Hrdy, A Course of Pure Mthemtics, Cmbridge University Press Wrehouse, London, 908. [2] T. M. Apostol, Mthemticl Anlysis, 2nd Edition, Person Eduction Asi Limited nd Chin Mchine Press, Beijing, [3] H. S. Nik, F. Soleymni, A Tylortype numericl method for solving nonliner ordinry differentil equtions, Alexndri Engineering Journl, 52 (203), [4] R. Brrio nd M. Rodríguez nd A. Abd nd F. Bles, Breking the limits: The Tylor series method, Applied Mthemtics nd Computtion, 27 (20), [5] G. Groz, M. Rzzghi, A Tylor series method for the solution of the liner initil boundryvlue problems for prtil differentil equtions, Computers nd Mthemtics with Applictions, 66 (203), [6] K. Mleknejd, N. Aghzdeh, M. Rbbni, Numericl solution of second kind fredholm integrl equtions system by using Tylorseries expnsion method, Applied Mthemtics nd Computtion, 75 (2006), [7] M. MsjedJmei, H. M. Srivstv, An integrl expnsion for nlytic functions bsed upon the reminder vlues of the Tylor series expnsions, em Applied Mthemtics Letters, 22 (2009), [8] M. R. Eslhchi, M. Dehghn, Appliction of Tylor series in obtining the orthogonl opertionl mtrix, em Computers nd Mthemtics with Applictions, 6 (20), Received: My 26, 204
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