Question

Power Series help. *question attached below* Will give medal

Answer 1

So I've completed the first two parts of this question, but I'm not sure how to tackle the last part with the power series? Can I have some assistance, please?

Answer 2

Show me the terms you got for part 1 and 2 and I'll show you how to use those to get part 3! Most of the work is actually already done! =D

Answer 3

Really all you need to do is substitute in the first few power series terms of sin(2x) into the first few power series terms for ln(1+2x) but you have to make a slight modification, that's all. Then you can kind of throw away all the higher power exponents when you expand the polynomials.

Answer 4

I got this for the first part

Answer 5

And I got this for the second part. I don't quite understand what you're explaining to do for the last part though :$

Answer 6

Here, I'll give an example of something simpler so you can see what I mean.

Let's say these are taylor series approximations: \[\Large f(x) \approx 3-x+5x^2 \\ \Large g(x) \approx 1+9x+ 2x^2\] So if you want to compute the first few terms of the approximation of \[\Large f(2g(x)) \] All we have to do is plug it in: \[\Large f(2g(x))=f(2+18x+4x^2)=\\ \Large 3-(2+18x+4x^2)+5(2+18x+4x^2)^2 \\ \Large messy \ algebra\]

Then you can see this gives us the first few terms as well, if you want the first three terms, notice that we can stop ourselves from doing a lot of computation. I hope that wasn't too confusing, I know there's a lot going on but that's sort of unavoidable with power series sometimes.

Answer 7

Hmm, yes I understand

Answer 8

Yeah please feel free to ask questions, I remember how I felt learning about taylor series. On one hand they're really kind of interesting and on the other they're a nightmare to compute.

Answer 9

Are my first and second part answers correct, by the way?

Answer 10

I ended up with 2x - 2x^2 + 4/3 x^3
Is this correct?

Answer 11

:) yes, it is.

http://www.wolframalpha.com/input/?i=Maclaurin+series+for+ln%28sin%282x%29%2B1%29

Answer 12

Yay! :D