Mathematica’s ability to symbolic evaluate derivatives to arbitrary order
D[Sin[x], {x, n}]
Sin[(n \[Pi])/2 + x]
and its ability to sum infinite series enables demonstrating convergence of example Taylor series
Sum[a^n/n! D[Sin[x], {x, n}], {n, 0, \[Infinity]}]
Sin[a + x]
another example
Sum[a^n/n! D[Exp[x], {x, n}], {n, 0, \[Infinity]}]
E^(a + x)