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Question:

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

is:

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.

s0ulr3aper07
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    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. – achille hui Nov 29 '18 at 09:43
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    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. – Barry Cipra Nov 29 '18 at 09:44
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    Use find $$\left(1+\dfrac13\right)^{-1/2}$$ – lab bhattacharjee Nov 29 '18 at 09:52
  • @labbhattacharjee I compared the terms in the binomial series and found the same result that you've stated. This yields D as the answer. Thanks for the help.

    I do have a question about this method though. When does it not hold? In other words, should I be checking for something before using it? Like convergence?

     – s0ulr3aper07 Nov 29 '18 at 10:24
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    @s0ulr3aper07, In https://math.stackexchange.com/questions/746388/calculating-1-frac13-frac1-cdot33-cdot6-frac1-cdot3-cdot53-cdot6-cdot/746975#746975, find https://en.wikipedia.org/wiki/Binomial_series#Convergence – lab bhattacharjee Nov 29 '18 at 10:35
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    Also keep in mind http://www.efunda.com/math/exp_log/series_exp.cfm – lab bhattacharjee Nov 29 '18 at 10:38

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