Welcome to week five of sequences and series.
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We're going to be looking at power series. These are series that look like this.
The sum, say, n goes from zero to infinity of
a sub n, just some numbers, times x to the n.
That means you get to pick a sequence a sub n.
For example maybe a sub n is 2 to the n, just the sequence of the powers
of two.
The important thing here is that a sub n doesn't depend on x in any way.
A sub n is just a formula in this case given in terms of n, but not x.
And from that sequence, you build the power series.
Well continuing with this example, if a sub n is 2 to the n, then the associated
power series is the sum n goes from zero to infinity of 2 to the n times
x to the n.
The cool thing is that in a ton of cases the power
series that we're building are actually
representing functions we already know about.
Well, what about this case?
What happens here?
Well, one over one minus x is the sum n goes from zero to infinity of x to the n.
That's just the formula for summing a geometric series.
Now look at what happens if I replace x by 2 to the n.
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1 over 1 minus 2x. That must be the sum n goes from
zero to inifinity of 2x to the n. Which is exactly what I've got here.
This is the sum n goes from zero to infinity of two to the n times x to the n.
So this mysterious seeming power series is actually just a complicated
way of writing down this very reasonable seeming function, 1 over
1 minus 2x.
The other cool thing is that power series are like polynomials.
If I just write down the first few terms, right, I
could just look at the first few terms of this series.
It's 1 plus 2x plus 4x squared plus 8x cubed.
If I truncate this series, I just get a polynomial.
And it is as soon as I write the plus
dot dot dot, as soon as I'm thinking of this as
sort of a polynomial that goes on forever,
well that's really what a, a power series is.
So power series are really very cool.
They let us translate a lot of our
intuition for polynomials into other, more complicated functions.
We're going to see that there are
power series representations from very complicated functions.
Like sine and cosine.
But, since these power series look like polynomials, its going to
let us translate some of that intuition
about polynomials into those more complicated transindental functions.
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