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

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Homework Help Overview

The problem involves finding the first four nonzero terms of the expression etcos(t). This falls under the subject area of series expansions and involves the multiplication of known series for the exponential and cosine functions.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the method of multiplying series terms and question the efficiency of this approach. There is a suggestion to use derivatives to find coefficients, with some participants expressing concern about the complexity of differentiating multiple times.

Discussion Status

The discussion is active, with participants exploring different methods for obtaining the series terms. Some guidance has been offered regarding the use of derivatives, though there is no explicit consensus on the best approach yet.

Contextual Notes

Participants express uncertainty about the feasibility of multiplying series directly and the challenges associated with differentiating the function multiple times. There is a recognition of the complexity involved in the problem setup.

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Homework Statement



Find the first 4 nonzero terms of:

[itex]e^{t}cos(t)[/itex]

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?
 
Last edited:
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1MileCrash said:

Homework Statement



Find the first 4 nonzero terms of:

[itex]e^{t}cos(t)[/itex]

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
 
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!
 
1MileCrash said:
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.
 
Mark44 said:
It wouldn't be that bad. Give it a try.

OK. Will report back.
 
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)
 

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