I'm working on a problem for university Calculus 2. We're talking about Taylor series right now and I need to approximate an integral using one of a function that I think it should be easy to produce a series for, but I'm not 100% sure. This is the function:
$$f(x) = 2x\sin(x)$$
I know the expansion for $\sin(x)$, which is in a reference table in the book. To give the first few terms it looks like this:
$$x – \frac{x^3}{3!} + \frac{x^5}{5!} – \frac{x^7}{7!} + \;\cdots$$
I'm pretty sure I can just multiply the whole polynomial by $2x$, giving:
$$2x^2 – \frac{2x^4}{3!} + \frac{2x^6}{5!} – \frac{2x^8}{7!} + \;\cdots$$
What's odd is that I can't find any examples quite like this in either of the textbooks or on the internet… which makes me wonder if this isn't actually a valid manipulation. Additionally, I can't seem to get an answer that matches this from Wolfram Alpha.
I could work out the Taylor series by hand, but the derivatives of the function start getting a bit ugly (by which I mean long), so I think I'm supposed to manipulate a known series since this should be an easy problem.
So, am I doing this right or am I on the wrong track?
Best Answer
Your approach is correct. WolframAlpha returns the same series for the following query:
Series[2 x Sin[x],{x,0,8}]