Find the sum of integers $a,b,c,d,$ and $e$ if $\dfrac{2011}{1990} = a+\dfrac{1}{b+\dfrac{1}{c+\dfrac{1}{d+\dfrac{1}{e}}}}$.
I could simplify the big fraction on the RHS, but I don't see how that would help. Also, there are infinitely many solutions to this equation so how should I find the integer solution?
$$\dfrac{2011}{1990}=1+\dfrac{1}{\dfrac{1990}{21}}=1+\dfrac{1}{94+\dfrac{1}{\dfrac{21}{16}}}$$
$$=1+\dfrac{1}{94+\dfrac{1}{1+\dfrac{1}{\dfrac{16}{5}}}}=1+\dfrac{1}{94+\dfrac{1}{1+\dfrac{1}{3+\dfrac{1}{5}}}}$$