WEBVTT
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In this segment, we will discuss
what a sequence is and
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what it means for a sequence
to converge.
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So a sequence is nothing but
some collection of elements
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that are coming out of some set,
and that collection of
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elements is indexed by
the natural numbers.
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We often use the notation, and
we say that we have a sequence
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ai, or sometimes we use the
notation that we have a
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sequence of this kind to
emphasize the fact that it's a
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sequence and not just
a single number.
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And what we mean by this is that
we have i, an index that
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runs over the natural numbers,
which is the set of positive
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integers, and each ai is
an element of some set.
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In many cases, the set is going
to be just the real
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line, in which case we're
dealing with a sequence of
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real numbers.
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But it is also possible that
the set over which our
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sequence takes values is
Euclidean space n-dimensional
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space, in which case
we're dealing with
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a sequence of vectors.
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But it also could be any
other kind of set.
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Now, the definition that
I gave you may
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still be a little vague.
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You may wonder how a
mathematician would define
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formally a sequence.
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Formally, what a sequence is,
is just a function that, to
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any natural number, associates
an element of S. In
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particular, if we evaluate the
function f at some argument i,
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this gives us the ith element
of the sequence.
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So that's what a sequence is.
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Now, about sequences, we
typically care whether a
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sequence converges to some
number a, and we often use
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this notation.
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But to make it more precise,
you also add
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this notation here.
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And we read this as saying
that as i converges to
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infinity, the sequence
ai converges to a
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certain number a.
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A more formal mathematical
notation would be the limit as
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i goes to infinity of ai is
equal to a certain number, a.
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But what exactly
does this mean?
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What does it mean for a
sequence to converge?
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What is the formal definition?
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It is as follows.
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Let us plot the sequence
as a function of i.
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So this is the i-axis, and here
we plot entries of ai.
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For a sequence to converge to a
certain number a, we need to
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the following to happen.
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If we draw a small band around
that number a, what we want is
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that the elements of the
sequence, as i increases,
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eventually get inside
this band and stay
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inside that band forever.
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Now, let us turn this into
a more precise statement.
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What we mean is the following.
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If I give you some positive
number epsilon, and I'm going
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to use that positive number
epsilon to define a band
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around the number a.
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So it's this band here.
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If I give you a positive number
epsilon, and therefore,
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this way, have defined a certain
band, there exists a
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time after which the entries
will get the inside the band.
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In this picture, it would
be this time.
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So there exists a time--
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let's call that time i0--
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so i0 is here such that after
that time, what we have is
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that the element of
the sequence is
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within epsilon of a.
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So this is the formal definition
of convergence of a
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sequence to a certain
number a.
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The definition may look
formidable and difficult to
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parse, but what it
says in plain
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English is pretty simple.
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No matter what kind of band
I take around my limit a,
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eventually, the sequence will
be inside this band and will
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stay inside there.
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Convergence of sequences has
some very nice properties that
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you're probably familiar with.
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For example, if I tell you
that a certain sequence
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converges to a number a and
another sequence converges to
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a number b, then we will have
that ai plus bi, which is a
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new sequence--
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the ith element of the sequence
is this sum--
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will converge to a plus b.
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Or similarly, ai times bi, which
is another sequence,
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converges to a times b.
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And if, in addition, g is a
continuous function, then g of
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ai will converge to g of a.
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So for example, if the ais
converge to a, then the
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sequence ai squared is going
to converge to a squared.