Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/238



We have so far discussed only a particular series (A) when the number of terms increases without limit. Let us now consider the general problem, using the series

whose terms may be either positive or negative. Denoting by $$S_n$$ the sum of the first $$n$$ terms, we have,

$$S_n = u_1 + u_2 + u_3 + \cdots + u_n$$, and $$S_n$$ is a function of $$n$$. If we now let the number of terms ($$= n$$) increase without limit, one of two things may happen: either


 * . $$S_n$$ approaches a limit, say $$u$$, indicated by

$$\lim_{n \to \infty}S_n = u$$; or


 * . $$S_n$$ approaches no limit.

In either case (C) is called an infinite series. In Case I the infinite series is said to be convergent and to converge to the value $$u$$, or to have the value $$u$$, or to have the sum $$u$$. The infinite geometric series discussed at the beginning of this section is an example of a convergent series, and it converges to the value 2. In fact, the simplest example of a convergent series is the infinite geometric series

$$a,\, ar,\, ar^2,\, ar^3,\, ar^4,\, \cdots,$$

where $$r$$ is numerically less than unity. The sum of the first $$n$$ terms of this series is, by 6, § 1,

$$S_n = \frac{a\left ( 1 - r^n \right )}{1 - r} = \frac{a}{1-r} - \frac{ar^n}{1-r}$$.

If we now suppose $$n$$ to increase without limit, the first fraction on the right-hand ,side remains unchanged, while the second approaches zero as a limit. Hence

$$\lim_{n \to \infty} S_n = \frac{a}{1 - r}$$,

a perfectly definite number in any given case.

In Case II the infinite series is said to be nonconvergent. Series under this head may be divided into two classes.

. Divergent series, in which the sum of $$n$$ terms increases indefinitely in numerical value as $$n$$ increases without limit; take for example the series

$$S_n = 1 + 2 + 3 + \cdots + n$$.