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Let $$P = 3^{\frac13}\times3^{\frac29}\times3^{\frac3{27}}\dots\infty$$

What would be the value of $P^{1/3}?$

$$P = 3^{1/3\ +\ 2/9 +\ 3/27\dots\infty}$$ $$Let\ S = \frac13 + \frac29 + \frac3{27}\dots\infty$$

How would I evaluate $S?$ I see 2 series in it, the one in the numerator, i.e, $1+2+3+4\dots$ and the one in the denominator $3+9+27\dots$ but can't proceed further.

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    $\begingroup$ Hint: Differentiate \begin{eqnarray*} \sum_{n=0}^{\infty} x^n = \frac{1}{1-x} \end{eqnarray*} $\endgroup$ Feb 13 at 14:07
  • $\begingroup$ @DonaldSplutterwit Is there a way to evaluate the series without differentiation or integration? $\endgroup$
    – Haider
    Feb 13 at 14:10
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    $\begingroup$ Hint: calculate $S-\frac13 S$ $\endgroup$
    – Blitzer
    Feb 13 at 14:12
  • $\begingroup$ This is an arithmetico-geometric series. $\endgroup$
    – peterwhy
    Feb 13 at 15:38
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    $\begingroup$ @peterwhy That link helped me, I didn't notice mine was a duplicate question. Thanks! $\endgroup$
    – Haider
    Feb 13 at 15:44

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