e^(t)cos(t), find first 4 terms centered at 0.
Alright, I know of two methods for doing this, and both seem remarkably painful.
I thought of listing 4 terms of each series (e^t and cost) and just multiplying them as polynomials and combining like terms. This wildly confuses me..
Then, someone recommended that I use derivatives to find the coefficients. Differentiating that function 4 times would be outright terrible.
Which way is the recommended?
I went with the second way, finding the fourth derivative wasn't nearly as bad as I thought.
First way seems wildly harder still. Is it a matter of opinion or would you say that I'm just doing the first way incorrectly if I think that?
well you don't want "the first four terms" of each series for the first method, you want the product of any terms that could contribute to the first 4 terms of the product.
so anything higher than a term of can be disregarded (but see below). if we write the coefficents of the terms for as and the terms for as , and the terms of as , then:
note that all the odd terms are 0, since cosine is an even function. we have:
so:
if we must calculate the first 4 non-zero terms, we need to calculate: