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The Taylor series for $e^x$ centered at 0 simplifies to sum $x^n/n$! because every derivative of $e^x$ is exactly $e^x$. When evaluating the nth derivative at the center $x=0$, the result is always $e^0$, which equals 1.

Conditions: $f(x)=e^x$; center $a=0$

The Taylor series for $e^x$ centered at 0 simplifies to sum $x^n/n$! because every derivative of $e^x$ is exactly $e^x$. When evaluating the nth derivative at the center $x=0$, the result is always $e^0$, which equals 1.

Conditions: $f(x)=e^x$; center $a=0$

A Maclaurin series is a special case of a Taylor series where the center of expansion is exactly 0. A Taylor series generalizes this by allowing the expansion to be centered at an arbitrary point a.

Conditions: The Taylor formula is taken with center a.; Setting $a=0$ recovers the displayed Maclaurin formula.

A Maclaurin series is a special case of a Taylor series where the center of expansion is exactly 0. A Taylor series generalizes this by allowing the expansion to be centered at an arbitrary point a.

Conditions: The Taylor formula is taken with center a.; Setting $a=0$ recovers the displayed Maclaurin formula.