Maclaurin series

[Mooch22]

The Maclaurin series for the function f is given by

f(x) = Summation when n=0 to positive infinity of (((2x)^(n+1))/(n+1)) = 2x + (4x^2)/2 + (8x^3)/3 +...+ on its interval of convergence.

a.) Find the interval of convergence of the Maclaurin series for f. Justify.
I have no idea what this means! HELP...please?


b.) Find the first four terms and the general term for the Maclaurin series for f'(x).
Do you just take the derivative of the function and plug in 0, 1, 2, and 3? and How do you take the derivative of (2x)^(n+1)??


c.) Use the Maclaurin series you found in part (b) to find the value of f'(-1/3)
If I find the derivative in part b, and just plug in (-1/3), shouldn't this work? If I could just get that derivative! :?: :?:


Thanks for the help![/code]

 

[pka]

\(\displaystyle \L
\begin{array}{l}
\sum\limits_{k = 0}^\infty {\frac{{\left( {2x} \right)^{k + 1} }}{{k + 1}}} \quad \Rightarrow \quad a_n = \frac{{\left( {2x} \right)^{n + 1} }}{{n + 1}} \\
\left| {\frac{{a_{n + 1} }}{{a_n }}} \right| = 2|x|\frac{{n + 1}}{{n + 2}} \to 2|x| < 1 \\
\end{array}\)
Solve for x to determine the interval of convergence (do not forget to check the endpoints)