Sequences

Theorem 8.1 Limits of Sequences from Limits of Functions

Suppose

is a function such that

for all positive integers n. If

, then the sequence

is also L

Assume the sequences

and

have limits A and B respectively. Then A)

where c is real number C)

provided B does not equal 0

Let r be a real number. Then

A) 0 if |r|<1 B) 1 if r=1 C) does not exist if r

1 or r>1 if r>0, then

is a monotonic sequence. If r<0, then

oscillates

Let

, and

be sequences with

for all integers n greater than some index N. If

. Then