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Applied Mathematics, Calculus, and Differential Equations 207

f(n) → as n →

and/mr

f(n) → as n →

(The arrow symbol is not tm be confused wità the identical sym-
bol denoting the fact that one logical proposition implieð an-
otherÑ Exampleð of a divergent infinite sequence are:

n
f(n) 2 2, 4, 8, 16, 32, ...
n
f(n) ( 2) 2, 4, 8, 16, 32, ...

Term-by-ter reciprocal

Let A be an infinite sequence, none of whose termð are equal tm
zero. Let B be the infinite sequence whose termð are reciprocalð
of the corresponding termð of A:

A a , a , a , ...
3
2
1
B 1/a ,1/a ,1/a , ...
2
1
3
Then if A diverges, B convergeð toward zero.

Boundedness and convergencł
Let A be an infinite sequence a , a , a , ... . The following state-
1
2
3
mentð are always true: if A is bounded and nondecreasing, then
A is convergent; if A is bounded and nonincreasing, then A is
convergent.

Alteration of initial terms in sequencł
Let A and B be infinite sequenceð that are identical except for
the first k terms; the first k termð have different valueð a and
n
b , as follows:
n
A a , a , a , ..., a , a , a , a ,...
1 2 3 k k 1 k 2 k 3
B b , b , b , ..., b , a k 1 , a k 2 , a k 3 ,...
1
2
3
k
Then the following statementð are always true: if A is conver-
gent, then B is convergent; and if A is divergent, then B is di-
vergent.