330 NEW SERIES for the 
in the firft place, that each term of our feries, taking its co-ef- 
ficient into account, is greater than one-fourth of the term be- 
fore it. 
aT 4 ii 
AGAIN, becaufe 7=3 (1+2”); and, fimilarly, v= (1+7”), 
and it having been proved that ¢’ <7, fo that fimilarly, ¢” <7’, 
abe a T a Seren hi 
therefore g (i 77), < z (1+7"), and confequently aro and 
re s ‘ 8 - 
i < ri Thus it appears, that each of the quantities 7’, &c. is 
lefs than a third proportional to the two immediately before it, 
and the fame muft alfo be true of the terms of the feries. 
60. Uron the whole, then, our firft feries for the calculation 
of logarithms may be expreffed as follows : 
5 Si getd Mthdy ee pik teetemeah tnobe ail on 
ogx  2%x%—I arcs ge ue seria 10,5 4 y 
+ Tm) + T(m+1) + R)3 
and here, as in the former feries, Tim) and T(m+1) denote any 
al 
: : : I 
two fucceeding terms, andR is a quantity greater than ~ T(m+1), 
3 
but lefs than a third proportional to T(m) — T(m41) and Tn+1) 3 
or it is lefs than 
Dcsusporis T(m41) — T(m) 
T (asta) 
3 (Lm) — Tomtx) PY 
Qobe 
Or. LHAT 
