Page 60 - Foundations Of Differential Calculus

2. On the Use of Differences in the Theory of Series 43
and

1
S.
(x + n)(x + n +1) (x + n +2) (x + n +3)
1 1
= − ·
3 (x + n +1) (x + n +2) (x + n +3)
1 1
+ · .
3 (n +1) (n +2) (n +3)

These forms can easily be continued as far as desired.
69. Since
1 1 1
S. = − ,
(x + n)(x + n +1) n +1 x + n +1

we also have
1 1 1 1
S. − S. = − .
x + n x + n +1 n +1 x + n +1
Although neither of these two partial sums can be expressed by itself, still
their difference is known. In many cases, by this means the sum of the series
can be reasonably found. This is the case if the general term is a quotient
whose denominator can be factored into linear factors. The whole quotient
is expressed as partial fractions, and then by means of this lemma it soon
becomes clear whether we can find the partial sum.
1
1
Example 1. Find the partial sum of the series 1+ + + 1 + 1 + 1 +··· ,
2 3 6 10 15 21
whose general term is 2/ x + x .
This general term can be expressed as
2 2
− .
x x +1
It follows that the partial sum is
1 1
2S. − 2S. .
x x +1
By the previous lemma we have that this is equal to
2 2x
2 − = .
x +1 x +1
Hence, if x = 4, then 8 =1 + 1 + 1 + 1 .
5 3 6 10
1 1 1 1 1
Example 2. Find the partial sum of the series , , , , ,..., whose
5 21 45 77 117
2
general term is 1/ 4x +4x − 3 .