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Demonstrating Convergence of Taylor Series

Posted 4 months ago
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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)
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