896 Summation of the Jlowly-converging 
2 . This obfervation fuggefts to us a method of finding 
a near value of the fum of this feries by the help of Sir 
Isaac newton’s binomial theorem, which may be ex- 
plained as follows. 
If m and n reprefent any two whole numbers, the re- 
ciprocal of the jth power of the binomial quantity 1 - x , 
or, according to Sir isaac newton’s notation of powers, 
- — m 
the quantity i-xl~, will , according to that celebrated 
theorem, be equal to the infinite feries 
m 
I + - X X 
n 
m m-f n 
4 — X 
n 
x XX -1 — x 
2 n n 2 n 
m ?n + n m-j -2 n 
X 
3 n 
x x- 
m m + n m-\- 2 n m+zn , 
4. _ X x x x X* 
n 2 n 3 « 4 n 
m m+n m-\~ 2 n m+zn m + 
+ - + x — — x — X — - - x X s + See. 
n 2 n z>n 4 ^ 5 n 
m-\-n 
2 n 
+ 6 n 
m + 2n m-j-zn m-\-A.n . 
CX 3 + DA? 4 + * 
3 n 4» S’ 2 
+ — ^fa 6 + ’^^■ gx ' 1 + H# 8 + 8ec. ; in which feries 
the capital letters a, b, c, d, e, f, g, h, See. ftand for i 
and the co-efficients of x, xx , a 3 , w 4 , X s , x 6 , x\ x s , 8ec. 
Now it is evident, that the generating fractions —— > 
’O+IH, ’H+V, >0+4?, m -pl, 2±^, ?+f n , Sec. are de- 
3 « 4 n $n 6 n 'jn on 
rived from - and from each other by the continual 
n J 
addition of n to both their numerators and denomi- 
nators. Therefore, though they are greater than they 
would be if m was fubtradted from the numerator 
of 
