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Taylor and Maclaurin series
A smooth function can often be approximated near a point by a polynomial built from the function's derivatives there. The resulting Taylor polynomial of degree $N$ around $c$ is
$$P_N(x)=\sum_{n=0}^{N}\frac{f^{(n)}(c)}{n!}(x-c)^n.$$
It matches the value of $f$ and its first $N$ derivatives at $x=c$.
Local approximation
For $e^x$ around $0$, successive Taylor polynomials use the beginning of the expansion
$$1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots.$$
Keeping only the first few terms gives a polynomial approximation near the center. More terms usually improve the approximation over a wider nearby region.
Taylor and Maclaurin series
Allowing the degree to grow without bound gives the Taylor series
$$\sum_{n=0}^{\infty}\frac{f^{(n)}(c)}{n!}(x-c)^n.$$
When $c=0$, it is called a Maclaurin series.
Convergence is not automatically equality
A Taylor series may converge without converging to the original function. Therefore the statement
$$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(c)}{n!}(x-c)^n$$
requires justification that the approximation error tends to zero.
For functions such as $e^x$, $\sin x$ and $\cos x$, the Taylor series does represent the function throughout its interval of convergence.
Taylor polynomials are useful even when the infinite-series question is not settled: their first purpose is local approximation, while the Taylor series asks whether that approximation becomes exact in the limit.