The 22nd Root Problem

Calculus Level 2

\(\begin{eqnarray} S_{1}&=&\sqrt[22]{1887}+\sqrt[22]{1888}+\cdots+\sqrt[22]{2013}+\sqrt[22]{2014} \\ \\ \ S_{2}&=&\sqrt[22]{1888}+\sqrt[22]{1889}+\cdots+\sqrt[22]{2014}+\sqrt[22]{2015} \\ \\ I&=&\int_{1887}^{2015}\!\sqrt[22]{x}\,\mathrm{d}x \end{eqnarray} \)

What can you say about the relative values of \(S_{1}\), \(S_{2}\), and \(I\)?

\[S_{1}<S_{2}<I\] \[S_{2}<S_{1}<I\] \[S_{2}<I<S_{1}\] \[S_{1}<I<S_{2}\] \[I<S_{1}<S_{2}\] \[I<S_{2}<S_{1}\]

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