Can someone explain to me how this equals? I'm taking a calculus III course at the moment, and I'm doing Taylor and Maclaurin series at the moment, and this is the last step of a problem, but i don't see how this equals each other (probably because I've never dealt with a sum times sum problem before or I can't recall doing one before anyways). If it at all matters, this was the original problem: f(x) = (sinx)ln(1+x). thanks.
Open up each parentheses and carry on the product grouping together equal powers of $\;x\;$ :
FYI, the first series is $\;\sin x\;$, which converges for all (real or complex) $\;x\;$ , whereas the second one is $\;\log(1+x)\;$ , which converges only for $\;|x|<1\;$