Maclaurin Series for eˣ

Calculus 2 · Axiom Academy

EXAMPLE Maclaurin Series for e^x Derive the exponential function as an infinite power series — then check how fast it converges. Find the Maclaurin series for f(x) = e^ x , then determine its radius of convergence. Radius of Convergence & Numerical Check Apply the ratio test to confirm where the series converges: Since the limit is for every x , the series converges for all real numbers. Radius of convergence: R = ∞. Numerical check — approximating by keeping terms up to degree n: The factorials in the denominator make the series converge fast — through degree 10 already matches e to five decimals. You derived and analyzed the Maclaurin series for e^ x . Here is what made it work: Special property: e^ x is its own derivative, so every f^ (n) (0) = 1 — that is what makes the derivation so clean. The series: , one of the most important series in mathematics. Universal convergence: the radius of convergence is , so it works for any real number x . Rapid convergence: the factorial denominators mean just a few terms give an excellent approximation. The same method — derivatives at 0 over n! — gives the Taylor series for , , and next.

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