Page 61 - Foundations Of Differential Calculus

44 2. On the Use of Differences in the Theory of Series
Since the denominator of the general term has the factors (2x − 1) and
(2x + 3), the general term can be expressed as
1 1 1 1 1 1 1 1
· − · = · − · .
4 2x − 1 4 2x +3 8 x − 1 8 x + 3
2 2
But then
1 1 1
S. 1 = S. 1 +2 − 1
x − x + x +
2 2 2
and
1 1 2 1
S. 1 = S. 3 + − 3 ,
x + x + 3 x +
2 2 2
so that
1 1 2 1 1
S. 1 − S. 3 =2 + − 1 − 3 .
x − x + 3 x + x +
2 2 2 2
When this is divided by eight, we have the desired partial sum:
1 1 1 1 x x x (4x +5)
+ − − = + = .
4 12 8x +4 8x +12 4x +2 3(4x +6) 3(2x +1) (2x +3)

70. General terms that have the form of binomial coefficients deserve
special notice. We will find the partial sums of series whose numerators are
1 and whose denominators are binomial coefficients. Hence, a series whose:

general term is has partial sum
1 · 2 2 2
− ,
x (x +1) 1 x +1
1 · 2 · 3 3 1 · 3
− ,
x (x +1) (x +2) 2 (x +1) (x +2)
1 · 2 · 3 · 4 4 1 · 2 · 4
− ,
x (x +1) (x +2) (x +3) 3 (x +1) (x +2) (x +3)
1 · 2 · 3 · 4 · 5 5 1 · 2 · 3 · 5
− ,
x (x +1) (x +2) (x +3) (x +4) 4 (x +1) (x +2) (x +3) (x +4)
and so forth. From this the law by which these expressions proceed is
obvious. However, the partial sum that corresponds to the general term
1/x is not obtained, since it cannot be expressed in finite form.

71. Since the partial sum has x terms if the index is x, it is clear that if
we let the index become infinite, we obtain the sum of these infinite series.