* invcfii gating the Sums cf infinite Series. 4 9 
lecling two terms Into one , beginning at the firjl term , the [urn of 
the femes thence arifing will be lefs than thefum of the given Jeries 
bf if*/* ries be formed by beginning at the Jecond Term , the 
fum thereof will be greater than the Jam of the given feries 
'by i-, 
2 r 
F or let — — — be any two fucceflive terms ofthe feries, 
rn-\-m n+a .r+m 
which, if we begin to collect at the firfl term (the firfl: term being 
4-) will be two terms to be colle&ed into one, and which will 
therefore give for a general term of the re- 
rn + m X n -f- a . r + tn 
fulting feries. Let us now make n infinite, and then the deno- 
minator of this term becomes infinite, and the numerator 
finite; therefore the terms of this latter feries at an infinite 
diftance becoming; infinitely fmall, the feries will there termi- 
nate. Now, by making n infinite in the given feries, the 
two fucceflive general terms at an infinite diftance become 
L - i ; confequently this feries is ftill continued after the other 
r r 1 
terminates ; and the terms of fuch a continuation will be 
fas they begin with — — - — by making n infinite) 
v jo rn-\-m ji + a.r + m 
1 — I + I — i. -f &c. which will alio be continued ad infill, and 
r r r r 
whofe fum by the lemma is ~ ; confequently the given feries 
exceeds that which is formed by collecting two terms into one, 
beginning at thefirfl, by ; hence the fum of the latter feries 
jl L will be equal to the fum of the former. If we begin to 
2 r 1 
Vol. LXXII. H h h collefl 
