i know how to do the first one the way where you make a table but is there any other way to do it?
these are the last two questions that i'll have for the rest of the semester (i hope)
find the taylor series for $f(x)=x^4-3x^2+1$ centered at the given value of $a=1$. assume that $f$ has a power series expansion. also find the associated radius of convergence.
find the sum of the series $\sum_{n=0}^{\infty} \frac{2^n}{3^nn!}$
I assume you mean a table of derivatives evaluated at $a$. For a general polynomial like this any way you come up with is going to boil down to evaluating $f^{(k)}(a)$.
For the 2nd problem you should know that
$e^x = \displaystyle{\sum_{n=0}^\infty}\dfrac{x^n}{n!}$
Thus
$\displaystyle{\sum_{n=0}^\infty}\dfrac{2^n}{3^n n!}=\displaystyle{\sum_{n=0}^\infty}\dfrac{(2/3)^n}{n!}=e^{2/3}$