The sum of $$1-\frac16+\frac16\times\frac14-\frac16\times\frac14\times\frac{5}{18}+\cdots$$


A) $\frac23$ B)$\frac{2}{\sqrt3}$ C)$\sqrt\frac23$ D)$\frac{\sqrt3}{2}$

After looking at the options I thought factoring into two's and three's would be a reasonable approach. The $\frac{5}{18}$ I considered factoring first as $\frac{2+3}{2\times3^2}$ and then as $\frac{3^2-2^2}{2\times3^2}$, but unfortunately I couldn't find a discernible pattern in either case. Any pointers in the right direction would be appreciated.

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    $\begingroup$ Hint: $\frac14 = \frac{3}{12}$. The $n$-th term is $(-1)^n\frac{(2n-1)!!}{6^n n!} = \left(-\frac{1}{12}\right)^n \binom{2n}{n}$. $\binom{2n}{n}$ is known as the central binomial coefficient, look at its wiki entry and you will know how to compute the sum. $\endgroup$ – achille hui Nov 29 '18 at 9:43
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    $\begingroup$ The pattern is completely unclear (to me, at least), but it does suggest an alternating sum of decreasing terms, from which we can infer that the answer is less than $1$ but greater than $1-{1\over6}={5\over6}=0.833333\ldots$. This rules out the first three options (e.g., $\sqrt{2/3}=0.816496\ldots$), leaving only $\sqrt3/2=0.866025\ldots$ as a possibly correct answer. $\endgroup$ – Barry Cipra Nov 29 '18 at 9:44
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    $\begingroup$ Use find $$\left(1+\dfrac13\right)^{-1/2}$$ $\endgroup$ – lab bhattacharjee Nov 29 '18 at 9:52
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    $\begingroup$ @s0ulr3aper07, In math.stackexchange.com/questions/746388/…, find en.wikipedia.org/wiki/Binomial_series#Convergence $\endgroup$ – lab bhattacharjee Nov 29 '18 at 10:35
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    $\begingroup$ Also keep in mind efunda.com/math/exp_log/series_exp.cfm $\endgroup$ – lab bhattacharjee Nov 29 '18 at 10:38

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