# Math Help - Finding terms for maclaurin series

1. ## Finding terms for maclaurin series

Problem- Find the terms through x^5 in the Maclaurin series for f(x)=e^(-x) * cos(x)

After trying to find f'(x), f''(x), f'''(x), and so on, the derivatives seemed overly complex. I heard that it is possible to use known Maclaurin series (e.g. e^(-x) and cos(x) separately) and then perform multiplications, divisions, etc. How would I combine them?

Thanks!

2. Originally Posted by dyy
Problem- Find the terms through x^5 in the Maclaurin series for f(x)=e^(-x) * cos(x)

After trying to find f'(x), f''(x), f'''(x), and so on, the derivatives seemed overly complex. I heard that it is possible to use known Maclaurin series (e.g. e^(-x) and cos(x) separately) and then perform multiplications, divisions, etc. How would I combine them?

Thanks!
Yes.

$e^x = 1 + x + \frac{x^2}{2!} + ...$
So,
$e^{-x} = 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!}+...$

And

$\cos x = 1 - \frac{x^2}{2!}+ \frac{x^4}{4!}-...$

That means,
$e^{-x} \cos x = \left( 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!}+... \right)\left( 1 - \frac{x^2}{2!}+ \frac{x^4}{4!}-... \right)$

Now preform something called the Cauchy Product.
On the first 5 terms in each one.

3. Originally Posted by dyy
Problem- Find the terms through x^5 in the Maclaurin series for f(x)=e^(-x) * cos(x)

After trying to find f'(x), f''(x), f'''(x), and so on, the derivatives seemed overly complex. I heard that it is possible to use known Maclaurin series (e.g. e^(-x) and cos(x) separately) and then perform multiplications, divisions, etc. How would I combine them?

Thanks!
Another way of doing this is to write:

$
f(x)= Re\left[ e^{x(-1+i)}\right]
$

Then:

$
f(x)= Re\left[
1 + [x(-1+i)] + \frac{[x(-1+i)]^2}{2!} + \frac{[x(-1+i)]^3}{3!}+\frac{[x(-1+i)]^4}{4!}+ ...
\right]
$

The right hand side of this is fairly easy to simplify to find the real part
and so find the required series.

RonL

4. I'm not sure I understand the cauchy product entirely unfortunately-

I'm trying to multiply the terms from the two series- so far I have-

1+(-1)x+(1/2-1/2)x^2+[the third term]<-where I'm stuck, since there's no term in cosx with x^3. What do I multiply the x^3/3! term from e^(-x) with, if anything?

Thanks once again.