Taylor series

The Maclaurin coefficients of $e^x$ are equal to $1/n!$ because every derivative of the exponential function is the function itself. When evaluating the $n$-th derivative at the expansion point $x=0$, the result is always $e^0 = 1$.

Conditions: The function is $f(x) = e^x$.; The expansion is centered at $x=0$ (Maclaurin series).; The general Taylor coefficient formula $c_n = f^{(n)}(a) / n!$ is applied.

Successive polynomial terms match higher-order derivatives at the expansion point $x=0$. The quadratic term matches the second derivative (concavity), the cubic term matches the third derivative, and so on.

Conditions: The function is $f(x) = e^x$.; The expansion is centered at $x=0$.; Terms are added sequentially from degree 2 to 4.

The local approximation is most visible in the close-up view near the expansion point $x=0$. In this region, the polynomial partial sums $S_n(x)$ hug the curve of $e^x$ tightly, demonstrating how matching derivatives at a single point creates a strong local fit.

Conditions: The animation includes a zoom or close-up feature.; The focus is on the neighborhood of the expansion point $x=0$.

While the Maclaurin series for $e^x$ converges at every fixed real $x$ and uniformly on each fixed bounded interval, this does not imply a uniformly small error over the entire real line. Furthermore, convergence does not guarantee that the error is strictly smaller at every point after every added term; the approximation quality can fluctuate locally depending on the degree and the specific $x$ value.

Conditions: The function is $f(x) = e^x$.; The series is the Maclaurin expansion centered at 0.; Considering pointwise and uniform convergence properties.

The degree-zero approximation $S_0(x)=1$ is a constant function that matches the value of $e^x$ at $x=0$ but carries no slope information. The degree-one approximation $S_1(x)=1+x$ adds the linear term $x$, which matches both the value and the first derivative (slope) at $x=0$, geometrically representing the tangent line to the curve at the expansion point.

Conditions: The function is $f(x) = e^x$.; The approximations are centered at $x=0$.; $S_0(x)$ is the constant term approximation.; $S_1(x)$ is the linear term approximation.