Page 49 - Foundations Of Differential Calculus

32 2. On the Use of Differences in the Theory of Series
51. It follows that a series of powers of the natural numbers will have
constant differences, as is clear from the following scheme:

First Powers: 1, 2, 3, 4, 5, 6, 7, 8, ...

First Differences: 1, 1, 1, 1, 1, 1, 1, ...
Second Powers: 1, 4, 9, 36, 49, 64, ...

First Differences: 3, 5, 7, 9, 11, 13, 15, ...

Second Differences: 2, 2, 2, 2, 2, 2, ...

Third Powers: 1, 8, 27, 64, 125, 216, 343, ...
First Differences: 7, 19, 37, 61, 91, 127, ...

Second Differences: 12, 18, 24, 30, 36, ...

Third Differences: 6, 6, 6, 6, ...

Fourth Powers: 1, 16, 81, 256, 625, 1296, 2401, ...
First Differences: 15, 65, 175, 369, 671, 1105, ...

Second Differences: 50, 110, 194, 302, 434, ...

Third Differences: 60, 84, 108, 132, ...

Fourth Differences: 24, 24, 24, ... .

The rules given in the previous chapter for finding differences of any order
can now be used to find the general terms for differences of any order for a
given series.
52. If the general term for any series is known, then not only can it be
used to find all of the terms, but it can also be used to reverse the order
and find terms with negative indices, by substituting negative values for x.
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For example, if the general term is x +3x /2 and we use both negative
and positive indices, we can continue the series in both ways as follows:
Indices: ..., −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, ...

Series: ..., 5, 2, 0, −1, −1, 0, 2, 5, 9, 14, 20, 27, ...

First Differences: ..., −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, 7, ...
Second Differences: ..., 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...