1. ## 2 more questions

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 $\sum_{n=0}^{\infty} \frac{2^n}{3^nn!}$

2. ## Re: 2 more questions

i know how to do the first one the way where you make a table but is there any other way to do it?

3. ## Re: 2 more questions

Originally Posted by tinspire
i know how to do the first one the way where you make a table but is there any other way to do it?
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

$\displaystyle{\sum_{n=0}^\infty}\dfrac{2^n}{3^n n!}=\displaystyle{\sum_{n=0}^\infty}\dfrac{(2/3)^n}{n!}=e^{2/3}$

4. ## Re: 2 more questions

i figured them out. thanks though. i just remembered that $e^x$ is equal to that.