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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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There are two criteria that are
the most commonly used for
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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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do not go down.
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In that case, those numbers may
go up forever without any
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bound, so if you look at any
particular value, there's
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going to be a time at which
the sequence has
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exceeded that value.
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In that case, we say
that the sequence
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converges to infinity.
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But if this is not the case,
if it does not converge to
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infinity, which means that
the entries of the
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sequence are bounded--
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they do not grow arbitrarily
large--
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then, in that case, it is
guaranteed that the sequence
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will converge to a
certain number.
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This is not something that we
will attempt to prove, but it
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is a useful fact to know.
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Another way of establishing
convergence is to derive some
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bound on the distance or our
sequence from the number that
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we suspect to be the limit.
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If that distance becomes smaller
and smaller, if we can
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manage to bound that distance
by a certain number and that
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number goes down to 0, then it
is guaranteed that since this
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distance goes down to 0,
that the sequence, ai,
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converges to a.
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And there's a variation of this
argument, which is the
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so-called sandwich argument,
and it goes as follows.
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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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that same number, a, and our
sequence is somewhere
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in-between, then our sequence
must also converge to that
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particular number, a.
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So these are the usual ways of
quickly saying something about
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the convergence of a given
sequence, and we will be often
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using this type of argument in
this class, but without making
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a big fuss about them, or
without even referring to
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these facts in an
explicit manner.