The Taylor polynomial of degree 100 for the function $f$ about $x=3$ is given by $P(x)=(x-3)^2 - \dfrac{(x-3)^4}{2!} + \dfrac{(x-3)^6}{3!}+ ... + (-1)^{n+1} \dfrac{(x-3)^{2n}}{n!} + ... - \dfrac{(x-3)^{100}}{50!}$. What is the value of $f^{(30)}(3)$ ?

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The coefficient for the 30^th derivative can be found with the 15^th term:

$$(-1)^{n+1} \frac{(x-3)^{2n}}{n!}$$

For $n=15$:

$$(-1)^{15+1} \frac{(x-3)^{2(15)}}{15!}$$ $$= \frac{(x-3)^{30}}{15!}$$

Taking the derivative with the power rule 30 times, results in:

$$= \frac{30\cdot 29\cdot 28...(x-3)^0}{15!}$$ $$= \boxed{\frac{30!}{15!} }$$