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Prove that the sum from 1 to ∞ of 2 / 3n-1 = 3

by RoRi

Prove that the following sum converges and has the given value.

    \[ \sum_{n=1}^{\infty} \frac{2}{3^{n-1}} = 3. \]

Here we use the formula for a geometric series,

    \begin{align*} \sum_{n=1}^{\infty} \frac{2}{3^{n-1}} &= \sum_{n=0}^{\infty} \frac{2}{3^n} &(\text{Reindexed sum}) \\[9pt] &= 2 \sum_{n=1}^{\infty} \left( \frac{1}{3} \right)^n \\[9pt] &= 2 \left( \frac{1}{1 - \frac{1}{3}} \right) \\[9pt] &= 3. \end{align*}

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