Multiplying Power Series: Help & Solutions

[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?

 

[dimensionless]

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"

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

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

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

 

[mkienbau]

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

 

[quantumdude]

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.

 

[mkienbau]

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