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Welcome to week six of Sequences and
Series.

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[MUSIC]

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Well thus far, we've been starting with a
power series, and

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then asking the question, what function
does that power series represent?

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For example, we considered this series,
the sum n goes from 0 to infinity of

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x to the n and we ended up showing that
this series is equal to 1 over 1 minus x.

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Provided that the absolute value of x is
less than 1.

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We've also built a lot of

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tools for transforming one power series
into a new power series.

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For instance, I can differentiate a power
series term by term.

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So in this case, what do I get?

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Well, if I differentiate this term by
term, I get the sum n

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goes from 1 to infinity of n times x to
the n minus 1.

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That's the derivative of x to the n with
respect to

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x, and that's the derivative of 1 over 1
minus x, and

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we calculated the derivative of 1 over 1
minus

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x, well, that's 1 over 1 minus x squared.

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So there we've got it, I've got a

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power series, and I found a function that
represents

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that power series at least on this
interval when

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the absolute value of x is less than 1.

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So that's what we've been doing, we've
been starting with the power series and

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we've been trying to identify, you know,

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what function does that power series
represent.

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And then maybe we'd be transforming

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those power series, messing around with
them by differentiating them term by

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term, integrating them term by term,

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multiplying them together, things like
that.

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But this week, we're going to turn all
that around.

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What I mean is that so far in the course,

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we've been starting with a power series
and then from

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that power series we've been getting an
iso description of

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the power series as some function that
we're familiar with.

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And, I want to turn that around now.

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I want to start with the description of a
nice function and then

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try to find a power series representing
that function on some interval.

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[SOUND]