Page 50 - Foundations Of Differential Calculus

2. On the Use of Differences in the Theory of Series 33
Since the general term is formed from the differences, any series can be
continued backwards so that if the differences finally are constant, the gen-
eral term can be expressed in finite form. If the differences are not finally
constant, then the general term requires an infinite expression. From the
general term we can also define terms whose indices are fractions, and this
gives an interpolation of the series.
53. After these remarks about the general terms of series we now turn to
the investigation of the sum, or general partial sum, of a series of any order.
Given any series the general partial sum is a function of x that is equal to
the sum of x terms of the series. Hence the general partial sum will be such
that if x = 1, then it will be equal to the first term of the series. If x =2,
then it gives the sum of the first two terms of the series; let x = 3, and
we have the sum of the first three terms, and so forth. Therefore, if from
a given series we form a new series whose first term is equal to the first
term of the given series, second term is the sum of the first two terms of
the given series, the third of the first three terms, and so forth, then this
new series is its partial sum series. The general term of this new series is
the general partial sum. Hence, finding the general partial sum brings us
back to finding the general term of a series.

54. Let the given series be
I II III IV V
a, a , a , a , a , a , ...
and let the series of partial sums be
I II III IV V
A, A , A , A , A , A , ....
From the definition we have
A = a,

I I
A = a + a ,
II I II
A = a + a + a ,
III I II III
A = a + a + a + a ,
IV I II III IV
A = a + a + a + a + a ,
V
I
V
II
A = a + a + a + a III + a IV + a ,
and so forth. Now, the series of differences of the series of partial sums is
I I II I II III II III
A − A = a , A − A = a , A − A = a , ...,
so that if we remove the first term of the given series, we have the series
of first differences of the series of partial sums. If we supply a zero as the