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Sequences
and limits |
The limit of a sequence |
The definition of the limit of a sequence |
Convergence of a sequence |
Verifying the convergence of a sequence from the definition, examples |
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The limit of a sequence |
The definition of the limit of a
sequence |
A
number L is
called the limit of a sequence {an}
if
for every positive number e there exist a natural number |
n0
such that if
n
> n0,
then |
an
-
L
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< e. |
That
is, a
number L is the limit of a sequence
if the distance between the term an
and L
becomes arbitrary small
by choosing n
large enough (see the examples above). The n0
denotes the value of the index n
starting from which the distance an
-
L
becomes smaller than the given e. |
Since the value of n0
depends on the size of e
it is usually written as n0(e). |
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Convergence of a sequence |
Therefore,
if a sequence {an}
has a limit L
we write |
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or |
an
®
L
as n
®
oo
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and
we say that that a sequence an
has the limit L
as n
tends to infinity. |
A
sequence {an}
is
convergent if it has a limit. Otherwise, we
say the sequence is divergent. |
Thus,
the sequences, (3),
(4)
and (5),
from the example above, all converge or tend to the limit 1. |
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Example: Let
examine the limit of the sequence (3)
given by |
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Solution:
Prove that |
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The
sequence |
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tends
to the limit 1
as the distance |
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can
become |
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arbitrary small by
choosing the natural number n
sufficiently large. |
For
example, if we choose
n = 100
the distance between an
and 1
is 1/n
= 0.01
that is, |
for
all n > 100
the distance | an
-
1 | < 0.01. |
Therefore,
n0(e)
= 101 meaning, starting from the 101st
term further, the distance of the remaining terms of the
sequence and 1,
is always less than 0.01. |
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Verifying
the convergence of a sequence from the definition, examples |
Example: Find
the limit of the sequence (2)
given by |
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Solution:
Let
prove that |
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Using
the definition of limit we must find a natural number
n0(e)
such that | an
-
0 | < e
for all n >
n0. |
Therefore,
if n >
n0 then
|1/n -
0 | = |1/n | = 1/n < 1/n0 < e. |
Suppose
we wish to make the difference (or the distance) between the anth
term and
the limit L
to be less than e
= 0.001 = 1/1000. |
Then,
as 1/n < e
or n
> 1/e
it follows n
> 1000
that is, starting from n0
= 1001
the distance an
-
L becomes smaller than the given e. |
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Observe
that the absolute value of terms of the sequences that converge to zero
become arbitrary small as n tends
to infinity that is,
| an
| < e
for all n >
n0(e). |
For
example such sequences are, |
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therefore |
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The
same way we can prove that the sequence (5)
above,
given by |
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converges
to 1. |
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As
can be seen on the number line above, the
terms of the sequence alternate from left to right approaching closer and closer to
1 as n
tends
to infinity. |
That
is, the sequence |
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alone alternately converges from left to right approaching closer and |
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closer to 0 as
n
tends to infinity. |
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Example: Find
the limit of the sequence,
0.9,
0.99, 0.999, . . . , 0.999 . . . 9 . . . , .
. .
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Solution: The
terms of the sequence can be written as
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Therefore,
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Example:
The
sequence
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has
the limit
3/2,
starting
from which term |
an -
L | < 0.01.
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Solution: Substitute
given values into the inequality |
an -
L
| < 0.01,
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Check the result by plugging
n0(e)
= 125
into |
an -
L | < 0.01.
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