Page 43 - Foundations Of Differential Calculus

26 2. On the Use of Differences in the Theory of Series
finally reduce to a constant; this chapter will be particularly concerned
with that class.
39. Two things are especially important concerning the nature of series:
the general term and the sum of the series. The general term is an expression
that contains each term of the series and for that reason is a function of the
variable x such that when x = 1 the first term of the series is obtained, the
second when we let x = 2, the third when x = 3, the fourth when x =4,
and so forth. When we know the general term of a series we can find any
of the terms, even if the law that relates one term to another is not clear.
Thus, for example, for x = 1000 we immediately know the thousandth
term. In the series

1, 6, 15, 28, 45, 66, 91, 120, ...
2
the general term is 2x − x.If x = 1, this formula gives the first term, 1;
when x = 2 we obtain the second term, 6; if we let x = 3, the third term
15 appears, and so forth. It is clear that for the 100th term we let x = 100,
and then 2 · 10000 − 100 = 19900 is the term.

40. Indices or exponents in a series are the numbers that indicate which
term we are concerned with; thus the index of the first term will be 1, that
of the second will be 2, of the third 3, and so forth. Thus the indices of any
series are usually written in the following way:

Indices 1 2 3 4 5 6 7 ...
Terms A B C D E F G ...

It is thus immediately clear that G is the seventh term of a given series.
From this we see that the general term is nothing else than the term of the
series whose index is the indefinite number x. First we will discover how
to find the general term of a series whose differences, either first, second,
or some other difference is constant. Then we will turn our attention to
finding the sum.
41. We begin with the first order, which contains arithmetic progressions,
whose first differences are constant. Let a be the first term of the series
and let the first term of the series of differences be b, which is equal to all
other terms of this series. Hence the series has the form:

Indices 1 2 3 4 5 6 ...
Terms a a + b a +2b a +3b a +4b a +5b ...
Differences b, b, b, b, b, b, ...

From this it is immediately clear that the term whose index is x will be
a +(x − 1) b and the general term will be bx + a − b. This is formed from