Finding Taylor Polynomial Approximations of Functions BC only

The $n$th-degree Taylor polynomial centred at $a$ is $\sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^{k}$, where the derivative is evaluated at the centre to give a number, the factorial cancels the one produced by differentiating $(x-a)^{k}$ down to a constant, and the power matches the order. Differentiating $P_n$ $k$ times and setting $x = a$ leaves exactly $f^{(k)}(a)$, which is what the factorial is for, so omitting it makes that derivative $k!$ times too large. The coefficient fails in two distinct ways: the factorial goes missing, or the derivative order is paired with a mismatched power, and the rule is that order, factorial and exponent are the same number in every term.

The centre is part of the object rather than a detail, so a polynomial built from $f^{(k)}(0)$ while written in powers of $(x-1)$ has halves referring to different points, and writing $x^{k}$ for $(x-a)^{k}$ is correct only in the Maclaurin case $a = 0$, whose tidiness is what disguises the general form. Two Taylor polynomials of one function at different centres are different polynomials agreeing with $f$ in different places, and neither is more correct. Drawn together, $\cos x$, $P_2$ and $P_4$ meet at the centre and separate as $|x|$ grows, with $P_4$ tracking the curve visibly longer: at $x = 2.5$ the values are $-0.8011$, $-0.4974$ and $-2.1250$. For $\cos x$ the odd derivatives vanish at $0$, so $P_2$ and $P_3$ are the same polynomial.