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I am watching this video on Probability, around the end there is a problem to be solved:

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I can't understand how a summation which extends from 1 to infinity can be equal to $\frac{1}{1-4/6}$

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The series $S = (\frac 46)^0\frac 16 +(\frac 46)^1\frac 16+... = ar^0 +ar^1 + ar^2...$ is a geometric series. You can search Geometric Series on internet to get the answer. In addition, you may get the result for the sum of the first n terms of a geometric series, which is $a \frac {1-r^n}{1-r}$ and also useful in statistics.

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You know the sum is finite, since it is a probability and so must be between $0$ and $1$. Let's call it $S$ where $$S=\frac16+\left(\frac46\right)\frac16 +\left(\frac46\right)^2\frac16 +\left(\frac46\right)^3\frac16 +\cdots$$

Multiplying through by $\frac46$ gives $$\frac46 S=\left(\frac46\right)\frac16 +\left(\frac46\right)^2\frac16 +\left(\frac46\right)^3\frac16+\left(\frac46\right)^4\frac16 +\cdots$$

and taking the difference gives $\dfrac26 S=\dfrac16$ i.e. $S=\dfrac12$, as you might intuitively expect from the symmetry in the question. This is a geometric series

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