IJRES Volue 3, Issue 8 (ugus 03) ISSN: 49-3905 MULIPLIERS FR HE - C,α SUMMBILIY F INFINIE SERIES Dr Meeashi Darolia* Mr Vishesh Basal** DEFINIINS ND NINS : Le = (a ) be a ifiie arix of coplex ubers a (, =,,...) ad le be a sequece of coplex ubers. Le a be a gie ifiie series wih sequece of parial sus (s ). We deoe by (s) he -rasfor of he sequece s=(s r ), s r a s (3.-) r he series a is suable, if r r s s ad i is said o be suable -, >, if r s s (3..3) whe = (resp. = as (resp., ) suabiliy., 0 ), he - suabiliy is he sae *ssisa Professor, Isharjyo Degree College Pehoa **ssisa Professor MIM Jagadhri Ieraioal Joural of Research i Egieerig & pplied Scieces 0 hp://www.euroasiapub.org
IJRES Volue 3, Issue 8 (ugus 03) ISSN: 49-3905 INRDUCIN: I 993, SULIMN proed he followig heore. HEREM : Le ( ) be a posiie o-decreasig sequece ad he sequece ( ) ad ( ) are such ha ad (3..) 0 as as, (3..) (3..3) (3..4) If here exiss o >0 such ha he sequece ( ) is o-icreasig Where, as (3..5) is he (C,l) ea of he sequece {a ), he he series a is suable C,,,. 3.3 Howeer, we shall proe he followig heore for C,, suabiliy. HEREM: Le ( ) be a posiie o-decreasig sequece ad he sequece ( ) ad ( ) are such ha he codiios (3..)-(3..4) of heore are saisfied. here exiss o >0 such ha he sequece ( posiie o-decreasig sequece such ha ) is o-icreasig ad ( ) is a, as (3.3.) where,as (3.3.) is he (C,l) ea of he sequece (a ), he he series a is Ieraioal Joural of Research i Egieerig & pplied Scieces hp://www.euroasiapub.org
IJRES Volue 3, Issue 8 (ugus 03) ISSN: 49-3905 Ieraioal Joural of Research i Egieerig & pplied Scieces hp://www.euroasiapub.org suable.,,, C 3.4 We eed he followig leas for he proof of our heore. Lea : MISHR ad SRIVSV If he codiio are saisfied, he ad Lea : SULIMN [80] If σ > δ > 0, he as ss PRF F HE HEREM : Le, be he -h (C,a) ea, 0<α<l, of he sequece a. he i order of proe he heore i is sufficie o show ha where, a! a... bels rasforaio gies,4,3,, say. o coplee he proof, i is sufficie, by Miowsis iequaliy o show
IJRES Volue 3, Issue 8 (ugus 03) ISSN: 49-3905 Ieraioal Joural of Research i Egieerig & pplied Scieces 3 hp://www.euroasiapub.org ha 4,,3, j, j, pplyig Holders iequaliy Now,, = (whe α=,, = 0, as 0 0,, as
IJRES Volue 3, Issue 8 (ugus 03) ISSN: 49-3905 Ieraioal Joural of Research i Egieerig & pplied Scieces 4 hp://www.euroasiapub.org =() i iew of ad he boudedess of. (BR) Moreoer,, φ =() gai,,3
IJRES Volue 3, Issue 8 (ugus 03) ISSN: 49-3905 Ieraioal Joural of Research i Egieerig & pplied Scieces 5 hp://www.euroasiapub.org =() I he iew of we hae = 0(), as i he case of x his coplees he proof of he heore.,4 +(), as i he case of, his coplees he proof of he heore. CRLLRY.
IJRES Volue 3, Issue 8 (ugus 03) ISSN: 49-3905 If we aig our heore reduces o heore.ss Ieraioal Joural of Research i Egieerig & pplied Scieces 6 hp://www.euroasiapub.org
IJRES Volue 3, Issue 8 (ugus 03) ISSN: 49-3905 REFERENCES. BLCI, M. : bsolue suabiliy facors; Co. Fac- Sci. Ui. ara, Ser., 9, (980), 63-68.. BR, H. : he absolue (p suabiliy facors of ifiie series; Porugaliae Mah., 45, (988), 3-37. 3. MZHR, S. M. C, suabiliy facors of ifiie series; ca Sci. Mah. Szeged, 7, (966), 67-70. 4. MISHR, B. P. : absolue Cesaro suabiliy facors of ifiie series; Red. Circl. Mah. Palero, 4, (965), 89-93. 5. MISHR, K. N.; SHRIVSV, R. S. L. : he absolue Cesaro suabiliy facors of ifiie series; Porugaliae Mah. Vol. 4, Fasc.,(983-984), 53-6. 6. SULIMN, W.. Mulipliers for he - C, suabiliy of ifiie series; aag J. of Mah., 4, (993), 95-0. Ieraioal Joural of Research i Egieerig & pplied Scieces 7 hp://www.euroasiapub.org