Page 62 - Foundations Of Differential Calculus
P. 62
2. On the Use of Differences in the Theory of Series 45
In this case, in the expressions just found, the later terms will vanish due
to the denominators becoming infinite.
Hence, these infinite series have finite sums, as follows:
1 1 1 1 2
1+ + + + + ··· = ,
3 6 10 15 1
1 1 1 1 3
1+ + + + + ··· = ,
4 10 20 35 2
1 1 1 1 4
1+ + + + + ··· = ,
5 15 35 70 3
1 1 1 1 5
1+ + + + + ··· = ,
6 21 56 126 4
1 1 1 1 6
1+ + + + + ··· = ,
7 28 84 210 5
and so forth. Therefore, every series whose partial sum we know can be
continued to infinity, and the sum can be exhibited by letting x = ∞,
provided that the sum is finite; this will be the case if in the partial sum
the power of x is the same in both the numerator and denominator.
