Find the Maclaurin series for e^x

A Maclaurin series is a Taylor series centered at zero. Its coefficients come from the derivatives of the function at x = 0.

Step 1: Write the Maclaurin formula

For a suitable function f, the series is f(x) = Σ[n=0 to ∞] f⁽ⁿ⁾(0)xⁿ/n!.

Step 2: Find the coefficients for e^x

Every derivative of e^x is e^x. At zero, each derivative equals e⁰ = 1. Thus the coefficient of xⁿ is 1/n!.

Final series

e^x = 1 + x + x²/2! + x³/3! + x⁴/4! + ⋯ = Σ[n=0 to ∞] xⁿ/n!.

The degree-four Maclaurin polynomial is 1 + x + x²/2 + x³/6 + x⁴/24. The infinite series converges for every real or complex value of x.