Not as complicated as it looks

Algebra Level 3

\[ \large \left ( \dfrac{\frac{f(1)}{g(1)}}{k(1)} \right )^{-1}+\left ( \dfrac{\frac{f(2)}{g(2)}}{k(2)} \right )^{-1}+\left ( \dfrac{\frac{f(3)}{g(3)}}{k(3)} \right )^{-1}+\cdots +\left ( \dfrac{\frac{f(100)}{g(100)}}{k(100)} \right )^{-1} \]

Let \(f(x)=1^{3}+2^{3}+3^{3}+\cdots +x^{3}\),
\(g(x)=1^{2}+2^{2}+3^{2}+\cdots+x^{2}\), and
\(k(x)=1+2+3+\cdots+x\).

Compute the expression above.

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