Here S is a function of n, and can be made as small as we please by increasing n; that is, the limit of S is 2 when n is infinite. This may be expressed by writing
lait : a. : :
a S=2. The sign = is sometimes used instead of the words “approaches as a limit.”
459. We shall often have occasion to deal with expressions consisting of a series of terms arranged according to powers of some common letter, such as
Clg + ® + Chg? + gt? + 2» + Ga oe
where the coefficients a, (4, Us, g++ are finite quantities independent of x, and the number of terms may be limited or unlimited.
It will therefore be convenient to discuss some propositions connected with the limiting values of such expressions under certain conditions.
460. Limiting Value. The limit of the series
(ty +t + ga? + 503 + oe
when x is indefinitely diminished is a_0.
(i.) Suppose that the series consists of an infinite number of terms.
Let b be the greatest of the coefficients a, dy, dg, +--+; and let us denote the given series by +S; then
S < bet be + be + 5
and if «<1, we have S< :
Thus when x is indefinitely diminished, S can be made as small as we please; hence the limit of the given series is a_0.
(ii.) If the series consists of a finite number of terms, S is less than in the case we have considered, hence still more is the proposition true.
