Sometimes it is very straightforward to determine whether the sum of a particular infinite series is rational or irrational. To take two examples, the sum of the series sum from k=1 to ∞ of 1/2~(km+n) is rational for any m, n∈ N , whilst that of 1 sum from k=1 to ∞ of 1/2~(F_k) is irrational, where F_k is the k th Fibonacci number. The former result may be shown to be true by utilising the formula for the sum to infinity of a geometric progression. The latter follows from the fact that the representation of a rational number as a bicimal, the binary equivalent of decimal, is necessarily eventually-periodic (which is clearly impossible in this case).
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