| Summation of Series. “ays 
tional, &c. quantities in the denominators of the fums may be 
eafily deduced from the preceding principles; and thence, by 
proceeding as is before taught, the fum required. 
The principles of all thefe cafes have been given in the 
Meditationes. 
8. Mr. James Bernovrii1 found fummable feriefes by 
affuming a feries V, whofe terms at an infinite diftance are 
_ infinitely little, and fubtraéting the feries diminithed by any 
number (/) of terms from the feries itfelf, &c. 
It is obferved in the Meditationes, that if T (mm), Tl’ (m +n), 
T(mtnta’), Timt+n+n' +n"), &e. be the terms at m, 
m+nz, m+n+tn', mint+n +n", &c. diftances from the firft, 
and aT (m)+4T (nn) eT (2 tu tn) + dT (m4n4n + 
nm’) +c. be the general term, it will be fummable, when 
a+6+c+d+&c,=0; the fum of the feries will be a(T(m) +T 
(m+1)4+T (m+2)-- 2... +T (m+n+n't0"+&e.—1)) 
+O(T (may tT (mtntry)4T (mtn+2)+ ... +T 
(m--n-en! +n! 4-&c.—1))- (Tn) +T(a+ntn’ +1) 
- 2. (Tmstnta’ +n’ +&e.—1))+&e.=H. If the fum 
a+btctd+é&c. be not=o, andthe feries T(m)+T(m-+r) 
+T(m--2)+ &c. iz infinitum be a converging one=S, then 
will the fum of the refulting feries be (4--61-c+d+&c.) 
S=- (64+c4+d4+&c.) (T*™ 2... +7") = (c+d+4+ &c.) 
aires Wott ea. (dt Bc. NC Lee et tt) 
Fe &c. 
8. 2. Let the feries V confift of terms, which have only one 
fa€tor in the denominator, and its numerator=1; that is, let 
the general term be ——, and the feries confequently 
7Z 
+e 
Stat Sa +é&c.= V3; from the before-mentioned addition 
b c wks 
or fubtraction there follows ep tae Weert; 4 &ei= 
F ff. aa" 
