31(3 PHILOSOPHICAL TRANSACTIONS. [ANNO 1782. 



at the 1st term, the series resolves itself into — — — — — - — — - — &c. and the 

 correction, to be added, being -, we have — — — — — g— — &c. + - for 



the sum of the given series. Now — — — ■ — - — - — -z~ — &c. is well known to 



° 2.34.50.7 



be equal to — 1 + hyp. log. of 2; consequently the sum of the given series is 

 = — i + hyp. log. of 2. 



If we begin to collect at the 2d term, the series becomes — — -\- (- - — ■ -f- 



&c. and the correction, to be subtracted, being 



- we have — - + — - -f — r- + &c. — - for the sum of the given series; but 



2 1.23.45.0 2 ° 



1 1- —p + &c. is equal to the hyp. log. of 2; therefore the sum of 



the given series is = — - + hyp. log. of 2, the same as before. 



Prop. 2. — If - — — be the general term of a series formed by writing: for n 



w+ nv ° JO 



any series of numbers in arithmetic progression, and whose terms are alternately 

 -f- and — ; then if a series be formed by collecting two terms into one, begin- 

 ning at the first term, the sum of the series thence arising will be less than the 

 sum of the given series by — . If a series be formed by beginning at the second 



term, the sum of it will be greater than the sum of the given series by — . 

 This is proved in a manner similar to the former. 



Exam. Let the given series be 1 1- &c. 



& 3 5 7 9 



Here x = 3, z = 2, w = I, v = 1, n = 2, 4, 6, 8, &c. Now, if we begin 



° 2 ° 



to collect at the first term, the series becomes — - + ■ -— • A — - 4- &c. and 



3.3 7 .y 11.13 



o <) 2 



the correction, to be added, being 1, we have — r -f- „\\ + 77 ... + &c. -|- ] 



for the sum of the given series; but if a = a circular arc of 45° whose radius 



2 2 2 



is unity, it is well known that — -|- — - -f -— ~ + &c. = 1 — A; therefore 



the sum of the given series is 2 — a. 



Besides the series contained in the foregoing propositions, a great variety of 

 other series might be produced where a correction is necessary, after collecting 

 two terms into one, in order to exhibit the true value of the given series. As 

 the proper correction however may always be found from the principles delivered 

 in the above propositions, that is, by considering what the terms of the given 

 series become at an infinite distance, Mr. V. only adds one or two instances more, 

 and concludes what he at present intended to offer on this subject. 



Exum. 1. Required to find the sum of the infinite series 

 3.4 4.5 . 5.o 6.7 , - 

 l.;-2-3+3-4-475 + &C - 



