LaTex error- !Paragraph ended before \split was complete

Question

I am using amsmath package but keep getting errors with the following:

\begin{split}
\[
A_R&=\frac{4}{9}+\frac{20}{81}+\frac{100}{729}+\frac{500}{6561}+\dots \\
&=\frac{4}{3^2}+5\times\frac{4}{3^4}+5\times5\times\frac{4}{3^6}+5\times5\times5\times\frac{4}{3^8}+\dots \\
&=\sum_{n=1}^{\infty}5^{n-1}\times\frac{4}{3^{2n}} \\
&=\sum_{n=0}^{\infty}5^{n-1}\times\frac{4}{3^2n+2}} \\
&=\frac{\frac{4}{9}}{1-\frac{5}{9}} \\ 
&=1
\]
\end{split}

The error is detected on the line after \end{split} and I cannot understand why. I also get:

! Argument of \split has an extra }.
! Extra }, or forgotten \endgroup.

Answer 1

The very last } of the line &=\sum_{n=0}^{\infty}5^{n-1}\times\frac{4}{3^2n+2}} \\ shouldn't be there. Plus, the split environment must be placed inside math mode, not outside:

\[
\begin{split}
A_R&=\frac{4}{9}+\frac{20}{81}+\frac{100}{729}+\frac{500}{6561}+\dots \\
&=\frac{4}{3^2}+5\times\frac{4}{3^4}+5\times5\times\frac{4}{3^6}+5\times5\times5\times\frac{4}{3^8}+\dots \\
&=\sum_{n=1}^{\infty}5^{n-1}\times\frac{4}{3^{2n}} \\
&=\sum_{n=0}^{\infty}5^{n-1}\times\frac{4}{3^2n+2}} \\
&=\frac{\frac{4}{9}}{1-\frac{5}{9}} \\ 
&=1
\end{split}
\]