Finding the First 4 Nonzero Terms of e^tcos(t)

[1MileCrash]

Homework Statement

Find the first 4 nonzero terms of:

$e^{t}cos(t)$

Homework Equations

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?

 

[Ray Vickson]

Homework Statement

Find the first 4 nonzero terms of:

$e^{t}cos(t)$

Homework Equations

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?

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

 

[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!

 

[Mark44]

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

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

 

[1MileCrash]

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

OK. Will report back.

 

[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)