documentclass{article}      % Specifies the document class

usepackage{amsmath, amsfonts, mathtools, xfrac, empheq}
usepackage{caption}


begin{document}       % End of preamble and beginning of text.
begin{tabular}{l r}
begin{minipage}{0.6textwidth}
begin{align}
    V^pi = E_pi { R_t mid s_t = s } \
    = E_pi left { sumlimits_{k=0}^infty gamma^k r_{t+k+1} mid s_t = s right }  notag \
    = E_pi left { r_{t+1} + gamma sum_{k=0}^infty gamma^k r_{t+k+2} mid s_t =s right} notag
end{align}
end{minipage}
&
begin{minipage}{0.4textwidth}
%includegraphics[width=textwidth]{olcar_L11-8}
captionof{figure}{Backup diagram}
end{minipage}
end{tabular}

end{document}

Overfull hbox (39.00003pt too wide) in paragraph at lines 8--22

documentclass{article}      % Specifies the document class

usepackage{amsmath, amsfonts, mathtools, xfrac, empheq}
usepackage{caption}


begin{document}       % End of preamble and beginning of text.

noindent
begin{minipage}{0.6textwidth}
begin{align}
    V^pi &= E_pi { R_t mid s_t = s } \
    &= E_pi left { sumlimits_{k=0}^infty gamma^k r_{t+k+1} mid s_t = s right }  notag \
    &= E_pi left { r_{t+1} + gamma sum_{k=0}^infty gamma^k r_{t+k+2} mid s_t =s right} notag
end{align}
end{minipage}hfill
begin{minipage}{0.4textwidth}
%includegraphics[width=textwidth]{olcar_L11-8}
captionof{figure}{Backup diagram}
end{minipage}


end{document}

documentclass{article}      % Specifies the document class

%usepackage{amsmath, amsfonts, mathtools, xfrac, empheq}
usepackage{amsmath, amsfonts, empheq}
usepackage{caption}


begin{document}       % End of preamble and beginning of text.
begin{tabular}{l r}
begin{minipage}{0.6textwidth}
$ % PS
%begin{align}
begin{aligned}
    V^pi &= E_pi { R_t mid s_t = s }         &= E_pi left { sumlimits_{k=0}^infty gamma^k r_{t+k+1} mid s_t = s right }  notag         &= E_pi left { r_{t+1} + gamma sum_{k=0}^infty gamma^k r_{t+k+2} mid s_t =s right} notag
%end{align}
end{aligned} $
end{minipage}
&
begin{minipage}{0.4textwidth}
%includegraphics[width=textwidth]{olcar_L11-8}
captionof{figure}{Backup diagram}
end{minipage}
end{tabular}

end{document}