Calculus Two: Sequences and Series is an introduction to sequences, infinite series, convergence tests, and Taylor series. The course emphasizes not just getting answers, but asking the question "why is this true?"

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From the course by The Ohio State University

Calculus Two: Sequences and Series

899 ratings

The Ohio State University

899 ratings

Calculus Two: Sequences and Series is an introduction to sequences, infinite series, convergence tests, and Taylor series. The course emphasizes not just getting answers, but asking the question "why is this true?"

From the lesson

Power Series

In this fifth module, we study power series. Up until now, we had been considering series one at a time; with power series, we are considering a whole family of series which depend on a parameter x. They are like polynomials, so they are easy to work with. And yet, lots of functions we care about, like e^x, can be represented as power series, so power series bring the relaxed atmosphere of polynomials to the trickier realm of functions like e^x.

- Jim Fowler, PhDProfessor

Mathematics

Welcome to week five of sequences and series. [MUSIC] 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 2x. [INAUDIBLE] 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. [SOUND]

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