1549901369-Elements_of_Real_Analysis__Denlinger_

498 Chapter 8 • Infinite Series of Real Numbers

Theorem 8.5.8 (Abel's Test) If L;ak converges and {bk} is a bounded,
monotone sequence, then I:; akbk converges.

Proof. Apply Dirichlet's test to the series L;(b - bk)ak or to the series

L;(bk - b)ak. (Work out the details in Exercise 11.) •

oo ( -1) k tan -^1 k
Example 8.5.9 The series I:; converges by Abel's test, since
k=l k
f: ( -l) k converges and {tan -^1 k} is monotone increasing and bounded above
k=l k
by ?T/2.

DOT PRODUCT OF SEQUENCES

In a linear algebra course, a "finite sequence" of real numbers,

is called an n-tuple or n-vector. We often use "vector" notation X! to denote

an n-tuple:

Recall that we add n-vectors X! and '[/ according to the rule

=(x1+Y1,x2+Y2,x3+y3,··· ,xn+Yn),

and we multiply an n-vector X! by a real number r (called a "scalar" in this

context) according to the rule

Examples 8.5.10 (a) (2, -4, 5, 0, -8) + (-1, 8, -9, 7, 0) = (1, 4, -4, 7, -8).

(b) -3(5, -2, 1, 0) = (-15, 6, -3, 0).

(c) 2(3, 4, -5) - 5(-2, 1, 7) = (6, 8, -10) + (10, -5, -35) = (16, 3, -45).

The set of all n-tuples of real numbers, together with the two algebraic
operations we have just defined, is called Euclidean n-space and is denoted
!Rn. The basic algebraic properties of !Rn are listed in the following theorem,
which is one of the fundamental results of elementary linear algebra.

Theorem 8.5.11 !Rn, together with the operations of addition and multiplica-

tion by scalars defined above, has the following properties:

1. 'v'r, '[/ E !Rn, X! + '[/ E !Rn.