How to evaluate $\frac13 + \frac29 + \frac3{27}\dots\infty$

Asked Feb 13 '24 at 14:01

Active Feb 13 '24 at 14:23

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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.

asked Feb 13 '24 at 14:01

Haider

1

Hint: Differentiate \begin{eqnarray} \sum_{n=0}^{\infty} x^n = \frac{1}{1-x} \end{eqnarray} – Donald Splutterwit Feb 13 '24 at 14:07
@DonaldSplutterwit Is there a way to evaluate the series without differentiation or integration? – Haider Feb 13 '24 at 14:10
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Hint: calculate $S-\frac13 S$ – Blitzer Feb 13 '24 at 14:12
This is an arithmetico-geometric series. – peterwhy Feb 13 '24 at 15:38
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@peterwhy That link helped me, I didn't notice mine was a duplicate question. Thanks! – Haider Feb 13 '24 at 15:44
You don't want to multiply by ∞ or add ∞ or raise a number to the ∞ power. (Three dots suffice to indicate what you mean.) – Dan Asimov Feb 13 '24 at 17:38
@DanAsimov Noted. – Haider Feb 14 '24 at 09:18

How to evaluate $\frac13 + \frac29 + \frac3{27}\dots\infty$

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