Welcome to week six of Sequences and Series.
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Well thus far, we've been starting with a power series, and
then asking the question, what function does that power series represent?
For example, we considered this series, the sum n goes from 0 to infinity of
x to the n and we ended up showing that this series is equal to 1 over 1 minus x.
Provided that the absolute value of x is less than 1.
We've also built a lot of
tools for transforming one power series into a new power series.
For instance, I can differentiate a power series term by term.
So in this case, what do I get?
Well, if I differentiate this term by term, I get the sum n
goes from 1 to infinity of n times x to the n minus 1.
That's the derivative of x to the n with respect to
x, and that's the derivative of 1 over 1 minus x, and
we calculated the derivative of 1 over 1 minus
x, well, that's 1 over 1 minus x squared.
So there we've got it, I've got a
power series, and I found a function that represents
that power series at least on this interval when
the absolute value of x is less than 1.
So that's what we've been doing, we've been starting with the power series and
we've been trying to identify, you know,
what function does that power series represent.
And then maybe we'd be transforming
those power series, messing around with them by differentiating them term by
term, integrating them term by term,
multiplying them together, things like that.
But this week, we're going to turn all that around.
What I mean is that so far in the course,
we've been starting with a power series and then from
that power series we've been getting an iso description of
the power series as some function that we're familiar with.
And, I want to turn that around now.
I want to start with the description of a nice function and then
try to find a power series representing that function on some interval.
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