Taylor Series

1. Nov 22, 2011

1MileCrash

1. The problem statement, all variables and given/known data

Find the first 4 nonzero terms of:

$e^{t}cos(t)$

2. Relevant equations

3. The attempt at a solution

I am trying to multiply the terms of two known series for my answer, but I'm not sure how to do it efficiently.

Should I list 4 terms of each series, then multiply them as if they were just normal polynomials?

EDIT, I tried doing that and it is completely unfeasible. What's the correct way?

Last edited: Nov 22, 2011
2. Nov 22, 2011

Ray Vickson

What is "unfeasible" about it? It is unpleasant, maybe, but perfectly feasible. It is not how I would do it, however: I would use derivatives to get the coefficients in the expansion.

RGV

3. Nov 22, 2011

1MileCrash

I see, so you would take the derivatives evaluated for 0? As the coefficients?

Should I always try that method first?

EDIT: Wait, why would you do it that way? Differentiating that 4 function 4 times would be terrible!

4. Nov 22, 2011

Staff: Mentor

It wouldn't be that bad. Give it a try.

5. Nov 22, 2011

1MileCrash

OK. Will report back.

6. Nov 22, 2011

1MileCrash

OK, it was definitely a lot easier than I anticipated. Thanks again.

1st:
e^t(cost - sint)

2nd:

-2e^t(sin t)

3rd
-2e^t(cos t + sin t)

4th

-4e^t(cos t)