Find the Maclaurin series of the function

Question

Find the Maclaurin series of the function: $f(x) = \int ^x_0 \frac{sin(t)}{t} dt$

Can anyone give me an idea on how to do this one? Thanks.

Answer 1

Recall that the Maclaurin series of a function $f$ is $$ f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n, $$ and specifically $$ \sin x = \sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!}. $$ So we have \begin{align} f(x) &= \int_0^x \frac{\sin t}t\ \mathsf dt\\ &= \int_0^x \sum_{k=0}^\infty \frac{(-1)^kt^{2k}}{(2k+1)!}\ \mathsf dt\\ &= \sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!}\int_0^x t^{2k}\ \mathsf dt\\ &= \sum_{k=0}^\infty \frac{(-1)^k x^{2k+1}}{(2k+1)(2k+1)!}, \end{align} where the interchange in summation and integration is justified by Fubini's theorem.