Answer by Kaind for Cannot understand calculations (Taylor's Theorem)

$\big( f^{i}(x) \frac{(x - t)^i}{i!} \big)' = f^{i+1}(x) \frac{(x - t)^i}{i!} + f^{i}(x) \frac{(x - t)^{i-1}}{(i-1)!}$ by product rule.

Sum over $i$, to see that the series is telescoping.