Infinite Sums and Products

The geometric series, Taylor’s series, and a number of examples previously discussed in this book, suggest that we may well study those limiting processes of analysis which involve the summation of infinite series from a more general point of view. In principle, any limiting value $$S = \mathop {\lim }\limits_{n \to \infty } {s_n}$$ can be written as an infinite series; we need only put $${a_n} = {s_n} - {s_{n - 1}}$$ for n > 1 and al = sl to obtain $${s_n}={a_1}+{a_2}+\cdots +{a_n},$$ and the value S thus appears as the limit of sn, the sum of n terms, as n increases. We express this fact by saying that S is the “sum of the infinite series” $${a_1} + {a_2} + {a_3} + \cdots $$ .