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4.7 Maclaurin’s Theorem

The special case $a=0\,$ of Taylor’s Theorem was known to Maclaurin, and is easier to remember.

Maclaurin’s Theorem

If $R_{n}(x)\rightarrow 0$ as $n\rightarrow\infty$ for $x$ near to $0$ , then function $f\,$ may be represented by its convergent Maclaurin series

f(x)=f(0)+f^{\prime}(0)x+{{f^{\prime\prime}(0)}\over{2!}}x^{2}+\dots+{{f^{(n)}% (0)}\over{n!}}x^{n}+\dots.

Note that $n$ appears together as the order of the derivative $f^{(n)}(0)$ , as the power $x^{n}$ and in $n!$ .