Find the function represented by the power series: $\sum _{m=0}^{\infty }\frac{z^m}{4^m}$

Find the function represented by the series $\sum _{m=1}^{\infty }\frac{(2x{)}^m}{m}$.

Consider the power series defined by $f\left(x\right)=\sum _{k=0}^{\infty }{x}^k=\frac{1}{1-x}$, for , and the partial sum of the first $n$ terms given by $S_n\left(x\right)=\sum _{k=0}^{n-1}{x}^k$. Define the remainder after $n$ terms as $R_n\left(x\right)=f\left(x\right)-S_n\left(x\right)$. Can the remainder be expressed as $R_n\left(x\right)=\frac{x^n}{1-x}$?

Find a power series representation centered at $0$ for $h\left(x\right)=\frac{4x}{(1+x^2{)}^3}$. Also, find the interval of convergence.

Determine the radius of convergence for the series: $\(\displaystyle\]\sum\)_{m=1}^{\(\infty\)} \(\frac{m!}{(3m)^m}\) z^m$