WEBVTT
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This will be a short tutorial
on infinite series, their
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definition and their
basic properties.
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What is an infinite series?
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We're given a sequence of
numbers ai, indexed by i,
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where i ranges from
1 to infinity.
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So it's an infinite sequence.
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And we want to add the terms
of that sequence together.
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We denote the resulting sum of
that infinity of terms using
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this notation.
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But what does that
mean exactly?
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What is the formal definition
of an infinite series?
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Well, the infinite series is
defined as the limit, as n
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goes to infinity, of the finite
series in which we add
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only the first n terms
in the series.
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However, this definition makes
sense only as long as the
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limit exists.
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This brings up the question,
when does this limit exist?
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The nicest case arises when
all the terms are
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non-negative.
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If all the terms are
non-negative,
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here's what's happening.
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We consider the partial sum
of the first n terms.
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And then we increase n.
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This means that we
add more terms.
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So the partial sum keeps
becoming bigger and bigger.
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The sequence of partial sums
is a monotonic sequence.
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Now monotonic sequences always
converge either to a finite
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number or to infinity.
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In either case, this
limit will exist.
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And therefore, the series
is well defined.
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The situation is more
complicated if the terms ai
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can have different signs.
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In that case, it's possible that
the limit does not exist.
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And so the series is
not well defined.
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The more interesting and
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complicated case is the following.
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It's possible that this
limit exists.
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However, if we rearrange the
terms in the sequence, we
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might get a different limit.
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When can we avoid those
complicated situations?
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We can avoid them if it turns
out that the sum of the
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absolute value of the numbers
sums to a finite number.
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Now this series that we have
here is an infinite series in
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which we add non-negative
numbers.
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And by the fact that we
mentioned earlier, this
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infinite series is always
well defined.
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And it's going to be either
finite or infinite.
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If it turns out to be finite,
then the original series is
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guaranteed to be well defined,
to have a finite limit when we
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define it that way, and
furthermore, that finite limit
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is the same even if we rearrange
the different terms,
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if we rearrange the sequence
with which we sum the
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different terms.