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So we looked at the formal
definition of what it means

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for a sequence to converge, but
as a practical matter, how

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can we tell whether a given
sequence converges or not?

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00:00:13,080 --> 00:00:16,309
There are two criteria that are
the most commonly used for

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00:00:16,309 --> 00:00:20,690
that purpose, and it's useful
to be aware of them.

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The first one deals with the
case where we have a sequence

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of numbers that keep increasing,
or at least, they

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00:00:28,500 --> 00:00:30,380
do not go down.

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00:00:30,380 --> 00:00:37,210
In that case, those numbers may
go up forever without any

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00:00:37,210 --> 00:00:41,180
bound, so if you look at any
particular value, there's

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00:00:41,180 --> 00:00:44,740
going to be a time at which
the sequence has

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00:00:44,740 --> 00:00:46,700
exceeded that value.

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00:00:46,700 --> 00:00:49,350
In that case, we say
that the sequence

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00:00:49,350 --> 00:00:51,680
converges to infinity.

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00:00:51,680 --> 00:00:55,810
But if this is not the case,
if it does not converge to

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00:00:55,810 --> 00:00:58,800
infinity, which means that
the entries of the

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00:00:58,800 --> 00:01:01,180
sequence are bounded--

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00:01:01,180 --> 00:01:04,230
they do not grow arbitrarily
large--

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00:01:04,230 --> 00:01:09,360
then, in that case, it is
guaranteed that the sequence

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00:01:09,360 --> 00:01:12,760
will converge to a
certain number.

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00:01:12,760 --> 00:01:16,180
This is not something that we
will attempt to prove, but it

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00:01:16,180 --> 00:01:19,900
is a useful fact to know.

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00:01:19,900 --> 00:01:24,580
Another way of establishing
convergence is to derive some

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00:01:24,580 --> 00:01:29,060
bound on the distance or our
sequence from the number that

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00:01:29,060 --> 00:01:32,440
we suspect to be the limit.

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00:01:32,440 --> 00:01:37,039
If that distance becomes smaller
and smaller, if we can

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00:01:37,039 --> 00:01:41,550
manage to bound that distance
by a certain number and that

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00:01:41,550 --> 00:01:47,789
number goes down to 0, then it
is guaranteed that since this

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00:01:47,789 --> 00:01:50,900
distance goes down to 0,
that the sequence, ai,

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00:01:50,900 --> 00:01:52,850
converges to a.

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00:01:52,850 --> 00:01:55,400
And there's a variation of this
argument, which is the

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00:01:55,400 --> 00:02:00,330
so-called sandwich argument,
and it goes as follows.

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00:02:00,330 --> 00:02:05,050
If we have a certain sequence
that converges to some number,

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00:02:05,050 --> 00:02:10,090
a, and we have another sequence
that converges to

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00:02:10,090 --> 00:02:15,070
that same number, a, and our
sequence is somewhere

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00:02:15,070 --> 00:02:20,750
in-between, then our sequence
must also converge to that

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00:02:20,750 --> 00:02:23,620
particular number, a.

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00:02:23,620 --> 00:02:29,040
So these are the usual ways of
quickly saying something about

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00:02:29,040 --> 00:02:32,970
the convergence of a given
sequence, and we will be often

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00:02:32,970 --> 00:02:37,160
using this type of argument in
this class, but without making

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00:02:37,160 --> 00:02:42,190
a big fuss about them, or
without even referring to

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00:02:42,190 --> 00:02:43,990
these facts in an
explicit manner.