How to find the power series expansion that converges to Fresnel integral?

Question

Fresnel integral is $S(x)=\int_{0}^x{\sin(t^2)\,dt}$.

I'm trying to see how the power series expansion for the integral is found , I have to tried to use Taylor Series for expanding $\sin(t^2)$ but i couldn't see a pattern in the derivations.

Answer 1

Taylor gives $$ \sin(t) = \sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)!}t^{2n+1} $$ applied at $t^2$: $$ \sin(t^2) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}t^{4n+2} $$ Integrated over $t$ from $0$ to $x$: $$ \int\limits_0^x \sin(t^2) \, dt = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!(4n+3)}x^{4n+3} $$

Answer 2

Hint: $$ \int_0^x\sin( t^2) dt=\frac12\int_0^{x^2}\sin y\frac{dy}{\sqrt y} =\frac12\int_0^{x^2}\sum_{n=0}^\infty(-1)^n\frac{y^{2n+1/2}}{(2n+1)!}dy. $$ The subsequent computation is obvious.