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Example 1
x
n
=1/n is decreasing. x
n
=
1+(−1)
n
2
is bounded since |x
n
| <
2. The sequence x
n
= n is unbounded.
If we let x
n
be the digit in the nth decimal place of the number
e, the Euler number (to be defined later in the text), then x
n
is a
well-defined sequence but it doesn’t have a simple defining equation.
Definition 7
Fibonacci sequence x
n
is defined recursively by the conditions
x
1
=1,x
2
=1,x
n
= x
n−1
+ x
n−2
,n 3.
Each term is the sum of the two preceding terms:
{1, 1, 2, 3, 5, 8, 13 ...}
This sequence arose when the 13th-century Italian mathematician
Fibonacci solved a problem concerning the breeding of rabbits.
Consider the sequence
n
n +1
. It appears that its terms are approaching
1 as n becomes large. In fact, the difference
1 − x
n
=1−
n
n +1
=
1
n +1
can be made as small as we like by taking n sufficiently large. Let
us indicate this by writing lim
n→∞
n
n +1
=1.
In general, the notation lim
n→∞
= L means that the terms of the
sequence x
n
approach L as n becomes large.
Definition 8
A sequence { x
n
} has the limit L and we write lim
n→∞
x
n
= L or
x
n
→ L as n →∞if we can make the terms x
n
as close to L as
we like by taking n sufficiently large. If lim
n→∞
x
n
exists, we say the
sequence converges (or is convergent). Otherwise, we say it diverges
(or is divergent).
4
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