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Designs and theoretical analysis are presented for a room temperature resonant-cavity-enhanced GeSn LED whose emission peaks at the 2 <inline-formula><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula>m wavelength. The Ge/GeSn/Ge PIN hetero-diode of length 1 <inline-formula><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula>m is embedded in a rib-type Ge-on-Si nanobeam having either 24 or 36 air holes. The maximum LED modulation bandwidth <inline-formula><tex-math notation="LaTeX">$f_{3\text{dB}}$</tex-math></inline-formula> is proportional to the Purcell factor and is inversely proportional to <inline-formula><tex-math notation="LaTeX">$\tau _{s\,p0}$</tex-math> </inline-formula> the GeSn bulk spontaneous emission lifetime. For an emission linewidth of 200 nm and <inline-formula> <tex-math notation="LaTeX">$\tau _{s\,p0}$</tex-math></inline-formula> of 10 ns, an <inline-formula> <tex-math notation="LaTeX">$f_{3\text{dB}}$</tex-math></inline-formula> of 1.6 GHz is predicted.