Multiplying Power Series: Help & Solutions

• mkienbau

mkienbau

How do I multiply power series?

Homework Statement

Find the power series:
$$e^x arctan(x)$$

Homework Equations

$$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!}$$

$$arctan(x) = 0 + x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7}$$

The Attempt at a Solution

So do I multiply 1 by 0, x by x and so forth? Or do I go 1 by 0, 1 by x? Or is there another way?

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You have to multiply 1 by the whole acrtan series, x by the whole arctan series, and so on. There might be a way to simplify it though. Wikipedia has this under "power series"

$$f(x)g(x) = \left(\sum_{n=0}^\infty a_n (x-c)^n\right)\left(\sum_{n=0}^\infty b_n (x-c)^n\right)$$

$$= \sum_{i=0}^\infty \sum_{j=0}^\infty a_i b_j (x-c)^{i+j}$$

$$= \sum_{n=0}^\infty \left(\sum_{i=0}^n a_i b_{n-i}\right) (x-c)^n$$

So I kind of treat it like F.O.I.L.?

Sort of. FOIL is the distrubutive law for (binomial)X(binomial). Here, you've got two infinitely long "polynomials". Obviously, you won't be able to write out all of the terms. :tongue:

Awesome, I think I got it, I only had to take it out to the $$x^5$$ term.