ABOUT THE CALCULUS OF SOME FAMOUS SEQUENCES. (n + 1)2

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1 ABOUT THE CALCULUS OF SOME FAMOUS SEQUENCES D. M. BĂTINEŢU-GIURGIU, DANIEL SITARU, NECULAI STANCIU Let be the sequeces L 2 ; I 2 ; B 2 ; L + +!!; I B ! 2!. I proble 579 published i Matheatical Gazette GM; vol. VI, , Trai Lalescu requires to copute L provig the that L e d i proble 2042 fro GM; vol. XIX; 93-94, Roeo T. Iculescu requires to copute I, provig the i the sae volue of GM that I. I GM; the year XCIV-989, D. M. Bătieţu-Giurgiu proposed the proble C:890, i which he requires the calculus of B d the he proved that B e. We ited to treat soe probles of this type, ore geeral. Let be r, s [0, d a ; b 0, such that: a 0, ; a r s b 0, I these coditios: 0. r a e r ; b s+ b s + Of course, accordig to Cauchy-D Alebert s theore we have: r r a r + d accordig to Cesaro-Stolz s theore we have: b s+ r + +r a +r a e r + s+ s+ s s + s+ s+ b s + I the followig we will preset geeralisatios for L 2 ; I 2 ; B 2. Theore 0.. If, p, s, r [0, ; a ; b 0, are so: the: 0.2 a 0, ; a r s b 0, + a + bp + + r+ps+p a b p r+ps+p s + p e r

2 2 D. M. BĂTINEŢU-GIURGIU, DANIEL SITARU, NECULAI STANCIU Proof. We have: a b p r+ps+p a r b p s+ b p s + p r s + p d if er u + a + bp + r+ps+p a b p + d also u + a + + r b+ p b the + + a r e ps+p u d so a b+ b a er a b+ r ps+p ; a 2 + b p u l u + a + r+ps+p+ + + r r+ps+p +r p + + er+ps+p + b+ b b + b b p b + b b p Therefore, ep eps+p b + b s s+ b s+ epb b eps+p eps+p e ps+p e + a + bp + a + r+ps+p b p r+ps+p a b p r+ps+p u a b p r+ps+p u l u l u a b p r+ps+p u l u l u s + p e r l e s + p e r

3 ABOUT THE CALCULUS OF SOME FAMOUS SEQUENCES 3 Theore 0.2. If, p, r, s [0, d a, b 0, are so: the: a 0, ; a r s b 0, + r+ps+p+ + a + bp + r+ps+p+ a b p 0.3 s + p e r Proof. Like i theore 0. we have: d v e r+ps+p+ But, r a e r ; r+ps+p a b p b b s + d s+ r s+ b p er s + p a s + p e r b + r+ps+p+ a If v b p + a + ; 2 bp + the + v d so r v l v b p r+ps+p+ + a + b + e r+ps+p+ + r+ps+p+ D + a + bp + r+ps+p+ a b p + a r a a e r e ps+p e r+ps+p+ a b p v r b b + p It follows: r+ps+p+ a b p r+ps+p a b p v l v v v l v l v ; 2 D s + p e r l e s + p e r

4 4 D. M. BĂTINEŢU-GIURGIU, DANIEL SITARU, NECULAI STANCIU. Applicatios Applicatio.. If i theore 0. we take r ; p 0; a! the d so: +! a! a + +!! L e Applicatio.2. If i theore 0. we take p r d a ; b the a b d so I Applicatio.3. If i theore 0. we take r, p 0, a 2!!; b, the b d so: a 2 +!! 2!! a + 2 +!! 2!! 2e Applicatio.4. If i theore 0.2 we cosider r, p 0, a!, b the a b d so: ! B e! Applicatio.5. If i theore 0.2 we take r, p 0, a 2!!; b the a 2, b d so !! 2 e 2!! 2 Applicatio.6. If i theore 0.2 we cosider r, p 0, a 2! 3!...! the: a + + a a +! + +! + + e d so, if x a, the: + 2 e 2 x + 2 x aely we have obtaied proble s 692 fro [6] result. More probles like the oes above c be foud i [3]. Refereces [] Bătieţu M. D., Sequeces. Albatros Publishig House, Bucharest, 979 [2] Bătieţu-Giurgiu M. D., Bătieţu-Giurgiu Maria, Bîrchi-Dai I., Seeescu Augusti, Matheatical Aalysis. MatrixRo Publishig House, 2003 [3] Bătieţu-Giurgiu M. D., Katrois Aastasios, Stciu Neculai Calculatig the its of soe real sequeces. MathProbles, 204 [4] Bătieţu-Giurgiu M. D., Lalescu Sequeces. RMT, r. -2, 985, pp [5] Bătieţu-Giurgiu Maria, O Lalescu Sequeces Octogo Matheatical Magazie., vol. 3, r. A, April 2005, pp

5 ABOUT THE CALCULUS OF SOME FAMOUS SEQUENCES 5 [6] Bătieţu-Giurgiu M. D., Stciu N., Proble 692, The Petago Joural., Fall 20, pp 54 [7] Bătieţu-Giurgiu M. D., Stciu N., Proble 75, The Petago Joural., Fall 202, pp 44 [8] Bătieţu-Giurgiu M. D., Stciu N., Several results of soe classes of sequeces, The Petago Joural. volue 73, r. 2, Sprig 204, pp 0 23

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