On summation of certain infinite series and sum of powers of square root of natural numbers

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1 Notes on Nuber Theory and Discrete Matheatics ISSN 0 5 Vol 0, 04, No, 6 44 On suation of certain infinite series and su of powers of square root of natural nubers Raesh Kuar Muthualai Departent of Matheatics, DG Vaishnav College, Arubaa Chennai-60006, Tail Nadu, India e-ail: rajan_80@yahooco Abstract: Suation of certain infinite series involving powers of square root of natural nubers is evaluated through Rieann zeta function The su of powers of square root of first n natural nubers are expressed in ters of infinite series and Rieann zeta function Keywords: Hurwitz zeta function, Rieann zeta function, Infinite series, Raanujan noteboos AMS Classification: M06, M5, 40A05 Introduction Raanujan studied suation of following infinite series ) for x 0 and odd integer s in his paper, titled Suation of a certain series, through Hurwitz zeta function and Rieann zeta function [] Φx, s) = = ) s ) ) s x Φx, s) and Φ0, s) can be expressed in ters of Hurwitz zeta function ζx, s) and Rieann zeta function ζs) respectively In [4], he gave soe finite expressions for su of powers of n natural nubers in ters of zeta function and Φ Soe of the are n = n n 5 n = 5 n 6 ) n 4π ζ n 8 ) 5 6π ζ Φ, n) ) 6 Φ5, n) ) 40

2 Further, he pointed that the higher powers such as n 5/, n 7/, are not so neat as ) and ) Siilar expressions for su of powers of square root of first n natural nubers are also found in [, p ] In the present wor, we study the suation of following infinite series for odd integer s greater than n and n N x ) n )) ) s We give soe finite expressions for su of powers of first n natural nubers in ters of above entioned infinite series and Rieann zeta function Moreover, we give soe elegant expressions for n 5/ and n 7/ in ters of single infinite series and Rieann zeta function as given in ) and ) Preliinaries Definition For n N, x 0 and s be an odd integer greater than n, then define Φn, x, s) = x ) n )) ) s ) Fro the definition, it is clear that { Φn, 0, s) = Φn,, s) If n N Φn,, s) If n = 0 ) Usually, the Hurwitz zeta function and Rieann zeta function [, p07] are defined by ζs, x) = x ) s and ζs) = s respectively The functional equation satisfied by ζs) [, p 08], viz ζ s) = π) s Γs)ζs) cos Further, we have following identities fro Ref [] { πs ) ) s s n s = ζ s) ζ s, n ) s s n ) s = s )ζ s) s ζ s, n ) 4) Let ft, a, b) = a b ) s t a b s t) 5) 7

3 Then, expand 5) by binoial theore and after siplification, gives s = s ) ) a s b =! t [ft, a, b)] 6) t= Main results Theore If s > and W, then Φ, x, s) converges for all x 0 Proof Fro the definition Φ, x, s) = x ) )) ) s Using the inequality x <, x 0 and W, gives Φ, x, s) < s ) x ) s Using binoial theore and definition of Hurwitz zeta function, yields Φ, x, s) < s ) s ) ζ n n, x ) The right hand side of ) exists if s > Hence, Φ, x, s) converges if s > Theore Let N and s be an odd integer If s > 4, then Φ, x, s ) = x x ) gx, s, ) ) s )/ s )! ) ζ s/, x) f )!, where f ) = a b x [fa, b, x)] ) a=,b=,x= gx, s, ) = ) x x ) s x x ) s 4) Proof For a, b R and x 0, define γa, b, x) = a x b ) s x a x b s x ) 5) a x b x ) s a x b x ) s 8

4 Using binoial theore, expand right hand of 5) in ascending powers of x and after siplification, that γa, b, x) = 4 Using 6) in 5) can be written as s, γa, b, x) = x s/ s, s = s ) ) a s b 6) f! xs/ t 7) t= Replacing x by x n and taing suation on both sides for n = 0,,, and after siplification, gives γa, b, x n) = s,! ζ s/, x) f t 8) t= Differentiate 8) ties partially with respect to a and b, setting a = and b =, then [γa, b, x n)] a b = a=,b= s, f ) ζ s/, x) 9)! where f ) = [fa, b, x)] a b x Now, differentiate 5) ties partially with a=,b=,x= respect to a and b, set a = and b =, then γx, a, b) s )! = a x b ) s x x x ) a b ) a x b ) s x x x ) a x b x ) s x x ) ) a x b x ) s x x ) Replace x by x n and then taing suation on both sides for n = 0,,, and using ), gives s )! γx n, a, b) a b = ) φ, x, s ) a=,b= x x ) / [ x x ] s ) [ x x ] s ) Using 8) in above equation and after siplification, gives ) 9

5 Theore If N and s > is an odd integer, then s )/ s )! Φ, 0, s ) = ), f ) A) 0)! where A) = ) π) s Γ s ) ζ s ) cos sπ 4 ) Proof Let x = in ) then s )/ s )! Φ, 0, s ) = ), Using ) and after siplification, gives 0) f ) ζ s/)! Theore 4 If N and s > 4 is an odd integer, then where ) ) s = C) s )! ) s ), ) s ) s ) f )! A) C) = i [ ) i) s i) s ] ) Proof Put x = / in ) and siplification, yields ) Theore 5 If n N and s > 4 is an odd integer, then s ), f )! S s n) = ) s )! Where S n) = n s ), f )! [ φ, n, s ) A) n s ) 4) n n ) ] gn, s, ) 40

6 Proof Put x = n in ), then Φ, n, s ) = n n ) gn, s, ) s )/ s )! ) ζ s/, n) f )! Using 4) and after siplification, we find, [ Φ, n, s ) n n ) ) gn, s, )] s )! = s )/,, After siplification, copletes the proof A) Ss/ n) n s/ ) f )! 4 Evaluation of certain infinite series 4 Suation of infinite series through ain results In this subsection, soe infinite series are expressed in finite ters using theores in section Exaple 4 Let = and s = 7, 9 in ) Then ) ) 4 = 05 8π ζ ) ) 6 = 945 8π 4 ζ Let = and s =, in 0) Then ) ) 5 = 095 π 5 ζ ) ) 7 = 55 π 6 ) 7 π ζ ) π ζ ) 945 π ζ ζ ) 745 6π ζ 4 ) 7 ) ) 5 ) 9 7 8π ζ ) 5 4

7 Exaple 4 Let = and s = 7, 9 in ) Then ) ) 4 = 054 π ζ ) ) 6 = π ζ 4 ) 7 π ζ ) ) π ζ ) 5 Raanujan shown that the higher powers of S 5 n) are not so neat as ) and ) In the following exaple, we express S 5 n) and S 7 n) in a single infinite series Exaple 4 Let s = 7 and = in 4) Then 5/ 5/ n 5/ = 5 64π ζ Let s = 9 and = in 4) Then 68 7/ 7/ n 7/ = π ζ n 04 4 Suation of other infinite series The following infinite series ) 7 7 n 7 n n 8 n n 5 n )n ) n n ) 5 ) 9 9 n n n 5 0n n )n ) n n ) 7 ) / ) s and x ) / ) s can be evaluated through the expression A n Φ, n, s n) Where A s are constant and Φ is still defined by ) The following are evaluation of these type of infinite series through Φ Exaple 44 Consider the following expression A x )) ) s 4) = Φ0, x, s ) A )Φ, x, s) 4

8 Let A = in 4) Then Φ0, x, s ) Φ, x, s) = x ) s 4) Let A = in 4) and replace s by s Then Φ0, x, s) Φ, x, s ) = x ) s 4) Adding and subtracting 4) and 4), then ) s 44) = [Φ0, x, s ) Φ, x, s) Φ0, x, s) Φ, x, s )] x ) s = 45) = [Φ0, x, s ) Φ, x, s) Φ0, x, s) Φ, x, s )] Exaple 45 Consider the following expression [ x ) ) ) s A x ))) ) s ] x )) B ) s = Φ0, x, s 4) A 4)Φ, x, s ) Let A = and B = in 4) Then Let A = and B = in 4) Then B A )Φ, x, s) x ) 5/ ) 5/ ) s 46) = Φ0, x, s 4) Φ, x, s ) Φ, x, s) ) 5/ x ) 5/ ) s 47) = Φ0, x, s ) 5Φ, x, s) 6Φ, x, s ) 4

9 Adding and subtracting 46) and 47), then ) 5/ ) s = [Φ0, x, s 4) Φ, x, s ) Φ, x, s) Φ0, x, s 4) 5Φ, x, s ) 6Φ, x, s)] x ) 5/ ) s = [Φ0, x, s 4) Φ, x, s ) Φ, x, s) Φ0, x, s ) 5Φ, x, s) 6Φ, x, 5 Conclusion The suation of certain infinite series Φ, 0, n) involving square root of powers of natural nubers has been studied in this paper The su of powers of square root of first n natural nubers has been expressed in ters of infinite series defined in ) and Rieann zeta function Soe nuerical exaples are given to evaluate suation of the infinite series Φ, 0, n) and su of powers of square root of first n natural nubers References [] Berndt, B C, Raanujan Noteboos, Part V, Springer Verlag, New Yor, 998 [] Gradshteyn, I S, I M Ryzhi, Tables of Integrals, Series and Products, 6 Ed, Acadeic Press, USA, 000 [] Raanujan, S, Suation of certain series, Mess Math, Vol XLIV, 95, [4] Raanujan, S, On the su of the square roots of the first n natural nubers, Jour Indian Math Soc, Vol VII, 95,