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Review question

When does $1 + 2/3 + (2/3)^2 + \dotsb$ first exceed $0.9999S_\infty$?
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Ref: R9185

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Question

Find the least number of terms of the geometric series \[\begin{equation*} 1 + \frac{2}{3} + \left( \frac{2}{3} \right)^2 + \dotsb \end{equation*}\]

that must be taken so that the sum of these terms exceeds \(99.99\%\) of the sum to infinity of the series.