Partial Sum Σ

9.3 PROPERTIES OF INFINITE SERIES

It is convenient to use capital sigmas, ∑, for partial sums and infinite series, as we did for finite and infinite Riemann sums. We write

for the mth partial sum,

for an infinite partial sum, and

for the infinite series. Thus S is the standard part of S_H,

Sometimes we start counting from zero instead of one. For example, the formula for the sum of a geometric series can be written

, where |c| < 1.

Infinite series are similar to definite integrals. Table 9.3.1 compares and contrasts the two notions.

The difference between them is that the infinite series is formed by adding up the terms of an infinite sequence, while the definite integral is formed by adding up the values of f(x)dx for x between a and b. The definite integral of a continuous

function always exists. But the problem of whether an improper integral converges is similar to the problem of whether an infinite series converges.