SERIES. 



will become, by making :r = — I, 



2.4 3-8 4- 16 5-32 



I I I I „ 



I + + +&C. 



2345 



-&c. = 



whence, by changing figns, we have 



I I 



'-V + 7 



1 &c. = hyp. log. 2 



4 5 



+ 



I 



+ 



+ 



16 



+ 



5 • 32 



+ &c. 



4-S 



- &c. 



= 0.5963473621237 



Euler alfo employed other methods for fumraable feries» 

 which we have not referred to either in the above article, or 

 in the articles iNXUEMENTsor Recurring Series, one of the 

 moft general of which is by means of certain fluxional ope- 

 rations ; but as this has been carried to a greater extent by 

 Lorgna, in his traft " de Seriebus convergentibus," we 

 (hall defer any further mention of it till we come to an ex- 

 planation of Lorgna's method. 



We ought to give here fome account of the differential 

 method of Maferes and Hutton, but our article having al- 

 ready been carried to a greater extent than is ufual for ma- 

 thematical fubjefts, we muft limit ourfelves to giving merely 

 the theorems, and leave the application of them to the inge- 

 nuity of the reader. 



8. Maferes dtjjferenhal Formula for Jlov/ly converging Series 



Let a -It- bx -y- c .v' -\- d x'^ ■{- &c. 



reprefent any feries, and D', D", D'", &c. the firll terms 

 of tlie fuccL-flive order of differences of the co-efficients 

 a,b, c, d, &c. which arefuppofcd continually to diminifh, then 

 will the fumof the above feries be exprelled by the differen- 

 tial feries 



D' 



D" x^ 

 (I + "Y 



&c. 



I -(- * ( I +• .*)* 

 which is necefiarily converging, provided x be equal to, or 

 greater than unity. By means of thij feries, the author 

 finds the circumference of the circle from the feries 



I I 



7 + 7 



I 



7 + 



- &c. 



true to feven places of decimals, by the fummation of ten 

 terms, whereas, in its original form, 10,000 of its terms will 

 only (jive two dec mals correal. For a further illuflration 

 of this method, the reader is referred to the Phil. Tranf. for 

 1775, or to the author's Trcatife on Converging Series. 



9. Hulloii's AJelhfid for f lowly converging Scries. — This me- 

 thod applies only to thofe feries whole terms are alternately 

 plus and minus, a»a— b + c — d+ Sec. the total fum of 

 which feries is given alternately in exccfs and defeft. 



by the fucceflive quantities 



ja — * 7a — 4 & 4- c 



2' 4 • 8 



iSa -lib + ^c- d ^1 a - 26 b + i6c - 6d + c 

 16 ' p ► 



&c.; each of thefe quantities, as we have ftated above, is an 

 approximation towards the whole fum ; the lird in excefs, 

 the fecond in defeft, the third in exccfs, and fo on ; but 

 each is a nearer approximation than the preceding. The 

 general formula for n terms is 



~[(2' - i)a- iA-n)b-(B- "-^^) 



which latter feries, though indefinite like the firft, is fo much 

 more conversjing, that 2j terms of it will give a refult as 

 true as 10,000 terms of the orif;inal feries. 



The fame formula is alfo applicable to certain diverging 

 feries, but we can only give fome of the moft remarkable 

 refults, as 



-( 



Q _ "(n — i) (n — 2) 



^)rf-&c.J 



The method of applying this formula to computation, 

 however, is fuch, that we mufl refer the reader for an ex- 

 planation of it to the author's Mifcellaneous Trads, pub- 

 lifhcd in 4to. in 1778, or to the new edition of the fame m 

 3 vols. 8vo. pubhfhed in 1812. 



10. Lorgna's Method of Series. — This confifls in multi- 

 plying the terms of the propofed feries by fuch powers of 

 an indeterminate quantity, that the fluxion of the whole 

 feries being taken, and then divided by x, there fhall refult 

 a known feries, from which the fum of the original one 

 may be readily derived. Thus, let there be propofed 

 the feries, 



1,1 I 



+ + -4- &C 



p + q p + 2q p + 33 

 Multiply each term fucceffively by 



X » > X » » « » , &C. J 



and there refults 



+ JL' + ±1 



+ &c. 



P + 9 P+2g p + iq 



Make the fum of this feries = S, and then taking the 

 fluxion on both tides, we have 



?S ' /_ + , r+i jt+j 



—r ■= X g -If X t +jr« + X 1 + Scc. 



X ' 



or -^— = I -j- ^. -}- x'- + .«•' -I- &C. = -1—: 



I — X 



whence S 



7(1 -X) 



; and confequently. 



XI -I- * ' -1- * ' 



/> + 9 /> + 2 ? /+77 



which, by making .v = i, become* the fame as the feries 

 originally propofed, viz. 



— ; — H + 1- -*- &c. 



/> + ? Z^+z? / + 3? /* + 4? 



It mufl be obferved, liowevcr, that in all fuch exprclTions, 

 the fluent mull be fo taken, as to vanifh when * = o, and 

 to be pcrfedlly integral when x = I. 



By a fimilar procefs, the author finds the fum of the 

 feries, 



I I I 



/> + ? P + '^q P + $9 



Oo 2 



/k-t-4? 



+ Sec. 



to 



