1. Taylor + Power Series

I was having some trouble with these:

1. Find the Taylor polynomial of degree 4 centered at c=2 for the function $\displaystyle f(x)=\sqrt[3]{x}$

2. Given $\displaystyle e \approx 1 + 1 + {1^2\over2!}+ {1^3\over3!}+ {1^4\over4!}+ {1^4\over4!}$, use Taylor's Theorem to obtain an upper bound for the error of the approximation. Then calculate the exact value of the error.

3. Find the radius of convergence of the power series $\displaystyle \sum_{n=0}^{\infty}{(2n)!x^{2n} \over {n!}}$

Thank you and sorry for all the questions!

2. Originally Posted by bakanagaijin
2. Given $\displaystyle e \approx 1 + 1 + {1^2\over2!}+ {1^3\over3!}+ {1^4\over4!}+ {1^4\over4!}$, use Taylor's Theorem to obtain an upper bound for the error of the approximation. Then calculate the exact value of the error.
Let $\displaystyle T_5(1)$ be the above Taylor polynomial.
We have $\displaystyle e^1 - T_5(1) = \frac{e^y}{6!}(1)^6$ where $\displaystyle 0<y<1$. Note that $\displaystyle e^y < 3$.
Thus, $\displaystyle |e - T_5(1)| \leq \frac{3}{6!}$.

3. Originally Posted by bakanagaijin
3. Find the radius of convergence of the power series $\displaystyle \sum_{n=0}^{\infty}{(2n)!x^{2n} \over {n!}}$
Ratio test,
$\displaystyle \frac{(2n+2)!|x|^{2n+2}}{(n+1)!}\cdot \frac{n!}{(2n)!|x|^{2n} } = (2n+1)|x|^2$
This never converges unless $\displaystyle x=0$.