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Sequences
and limits
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The
definition of the real number
e
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use the monotonic sequence theorem to prove that the sequence defined
by |
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is
increasing and its terms remain less than one fixed number as n
tends to infinity, that is, the sequence converge to the
number e. |
| Recall
the binomial expansion theorem |
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| Let
expand an
using the above theorem |
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Then,
to prove an+1
> an, that is,
that given is increasing sequence, we can expand an+1
the same way. |
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Observe
that, while passing from n
to n
+ 1, the an+1
expression gets one new positive term and, at the same
time, all the differences in parentheses raised as every subtrahend
decreased (i.e., denominators increased
to n
+ 1). Therefore, since an+1
> an the
sequence is increasing. |
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To
show that all terms of the sequence are less than a fixed number M
we will evaluate the quantity an
on a suitable
way. |
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If
we omit second term (consisting 1/n)
in every parenthesis,
they increase, hence
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If
instead of the factors, 3,
4,
. . . , n
in the denominators, starting from the second, we substitute 2,
the denominators
decreased so the fractions increased, thus |
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since
applied is formula for the sum of the finite geometric series whose
ratio is 1/2
and the first term 1.
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As
the right side still depends on n,
to get an expression independent of n,
the one which holds to all terms, drop
the term 1/2n
in the numerator of the right side, what increases the numerator, such
that |
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Therefore,
as all terms of the sequence are less than M = 3 then, the sequence has
a limit that is not greater
than 3. Let
write few terms of the sequence to show how slowly it increases,
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we finally write |
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The
limit of
sequence
theorems
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The
cluster point or accumulation point
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Every
sequence of numbers does not have to tend to a limit value. For example terms of the sequence, |
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shown
on the number line, |
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accumulate around two points, the odd terms
around 0 while the even terms accumulate around 2. Therefore, it
is the divergent sequence. |
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cluster point is such a point of a sequence in every however small
neighborhood of which lie infinitely many
terms of the sequence. In the example above such points are 0 and 2. |
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to this definition every limit point is the cluster point but inverse
does not hold. |
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is, a convergent sequence can have at most one cluster point, hence if a
sequence converges, then its limit
is unique. |
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Divergent
sequences
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sequence is convergent if it has a limit, otherwise it is divergent. A
divergent sequence has no a finite limit. |
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sequence of the natural numbers 1,
2, 3, . . .
, n, . . .
has
no a cluster point since it tends (or diverge) to
infinity,
so we write |
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| Generally, if
terms of a sequence {an}
can become greater than every arbitrary large natural number N, that is,
if for every given number N
there exists an index n0
such that |
| an
> N
for all n >
n0
(N), |
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say that the sequence {an}
tends to infinity or diverge to +
oo
and we write either |
| an
®
+ oo
as
n
®
oo or |
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| If
terms of a sequence {an}
are such that for every given negative number -
N,
of arbitrary large absolute value,
there exists an index n0
such that |
| an
> -
N
for all n >
n0
(N), |
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say that the sequence {an}
tends to the negative infinity or diverge to -
oo
and we write either |
| an
®
-
oo
as n
® oo
or |
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| For
example, the
sequence {-
n}
tends to negative infinity or diverge to -
oo
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Contents
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Copyright
© 2004 - 2020, Nabla Ltd. All rights reserved.
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