Finding Taylor Polynomial for eˣ

Real Analysis · Axiom Academy

EXAMPLE Finding the Taylor Polynomial for e^x Building the Maclaurin polynomial of e^x at a = 0 , then watching the approximation tighten. Find the Taylor (Maclaurin) polynomial of degree n for f(x) = e^x centered at a = 0 . Use it to write the degree- 4 polynomial explicitly, then identify the full Taylor series. Each Taylor polynomial agrees with e^x at x = 0 and tracks it on a wider interval as the degree grows. Near 0 the curves are nearly indistinguishable. Nice work — you derived the Taylor polynomial for e^x from its defining property and saw why the fit improves with degree. e^x is its own derivative: every derivative f^ (k) (x) = e^x , so at the center f^ (k) (0) = e^0 = 1 for all . Clean coefficients: the k -th coefficient is , giving . Degree 4 explicitly: , and versus . The full series: letting gives , valid for every real x (radius of convergence ). This series for e^x is one of the most useful in mathematics — it reappears throughout calculus, differential equations, and complex analysis.

This is the written version of the interactive lesson above. See the full Real Analysis course.