Obtaining Maclaurin series of different functions from simpler ones

Question

I've noticed that it's tedious or time consuming to calculate Maclaurin series of some functions like $e^{x^{2}}$ or $\sin(1/x)$ directly. But the use of the simple expansion of $e^{x}$ or $\sin(x)$ and substitution of $x^{2}$ or $\frac{1}{x}$ respectively for $x$ would yield the desired results. I know that this works for all functions however I'm looking for a justification or theory behind this.

Answer 1

For a convergent entire series

$$f(x)=\sum_{k=0}^\infty a_kx^k,$$

you can formally substitute a function for $x$ and write

$$f(g(t))=\sum_{k=0}^\infty a_kg^k(t),$$

which is valid as long as $g(t)$ lies in the domain of convergence of $f$. In particular,

$$f(t^m)=\sum_{k=0}^\infty a_kt^{km}.$$

You can also substitute a polynomial $p(t)$, and you still get an entire series. Beware anyway that if you regroup the terms, you may need absolute convergence to hold.

Now the entire series can be seen as a Taylor development, and by uniqueness of this development,

$$\sum_{k=0}^\infty a_kp^k(t)$$ is the Taylor development of $f(p(t))$.