I want to find out what the Taylor expansion of
$$F(x) = \int_0^x \frac{\sin(t)}{t} dt .$$
Am I wrong in saying that by the fundamental theorem of calculus, $F'(x) = sin(t)/t$? Should I continue from there? It just doesn't sit well with me for some reason.
Thanks for your time.
Best Answer
As justified by Fantini, your idea is good and you just continue. I supposed that you started using the Taylor series for $sin(t)$, divided by $t$, integrate between $0$ and $x$ and you are done. I suppose you arrived to something looking like $$x-\frac{x^3}{18}+\frac{x^5}{600}-\frac{x^7}{35280}+\frac{x^9}{3265920}-\frac{x^{11}}{ 439084800}+O\left(x^{13}\right)$$