The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages.


Sums of Infinite Series

The sum of an infinite series will be a real number which is close to the nth partial sum for large n, and infinitely close to the infinite partial sums. Before stating the definition precisely, let us examine some infinite series and their partial sum sequences, and guess at their sums.

Table 9.2.1

Series

Partial sums

Sum

1 + 0.1 +0.01 + 0.001 + ...

1, 1.1, 1.11, 1.111,...

10/9

1+ 1/2 +1/4 +1/8 +1/16 + ...

2

1 - 1 + 1 - 1 + 1 - 1 + ...

1, 0, 1, 0, 1, 0,...

?

1 + 1 + 1 + 1 + 1 + ...

1, 2, 3, 4, 5,...

1 + 1/2 + 1/3 + 1/4 + 1/5 + ...

1, 3/2, 11/6, 25/12, 137/60

?

3 + 0.1 + 0.04 + 0.001 + ...

3,3.1,3.14,3.141,...

π

DEFINITION

The sum of an infinite series is defined as the limit of the sequence of partial sums if the limit exists,

a1 + a2 + ... + an + ... = limn→∞ (a1 + ... + an).

The series is said to converge to a real number S, diverge, or diverge to ∞, if the sequence of partial sums converges to S, diverges, or diverges to ∞, respectively.

The sum of an infinite series can often be found by looking at the infinite partial sums a1 + ... + aH. Corresponding to our working rules for limits of sequences, we have the following rules for sums of series.

(1) If the value of every infinite partial sum is finite with standard part S, then the series converges to S,

a1 + ... + an + ... = S.

(2) If there are two infinite partial sums which are not infinitely close to each other, the series diverges.

(3)     If there is an infinite partial sum whose value is infinite, then the series diverges.

(4)      If all infinite partial sums have positive infinite values, the series diverges to ∞,

a1 + ... + an + ... = ∞.

Given an infinite series, we often wish to answer two questions. Does the series converge? What is the sum of the series?


Last Update: 2006-11-07