How can I do #20 without going through each option and doing out the series?

Question

anonymous · Answer

Attached is a picture of the problem

whpalmer4 · Answer

Well, you'd need to be familiar with the series expansion for the building blocks you've got there in those expressions.  Do you know the expansion for e^x or e^(x^2)?  How about sin x and cos x?  Do any of them look similar in form to the mystery function?

anonymous · Answer

I know that e^x is 1 + x/2 + x^2/6 ...etc. right? is that what you mean by building blocks?

whpalmer4 · Answer

Indeed!  Now what if you multiplied that series by x^2?  Doesn't it look like the mystery function, except with a pesky x^2 and x^3 term at the front?  By the way, isn't it 1 + x + x^2/2 + x^3/6 + x^4/24 ...?

anonymous · Answer

yeah it is, I forgot the x. hmm i have a feeling it is choice D

whpalmer4 · Answer

That's an excellent feeling :-)  Multiplying the sequence gets you x^2 + x^3 + x^4/2 + x^5/6 (= x^5/3!) + x^6/24 (=x^6/4!) etc....then subtract x^2 and x^3 and there's your mystery function.

anonymous · Answer

perfect thanks :). could you remind me what the building blocks are for sin(x) and cos(x)?