XL OM - I - Task 6

Calculate the sum of the series $ \sum_{n\in A}\frac{1}{2^n} $, where the summation runs over the set $ A $ of all natural numbers not divisible by 2, 3, 5.
A natural number $ n $ is not divisible by $ 2 $, $ 3 $, and $ 5 $ if and only if, when divided by 30, it gives one of the following eight remainders:






Therefore, the sum we need to calculate is equal to






where $ s_i $ denotes the sum of an analogous series formed from terms corresponding to those values of $ n $ that have the form






Thus,






(The correctness of the applied transformation and the validity of equation (1) follow from the fact that the terms of the considered series are positive; they can therefore be grouped and rearranged in any way without affecting the convergence and value of the series sum.)
The right-hand side of equation (2) represents a geometric series with a common ratio of $ 2^{-30} $. Its sum is equal to






Substituting these numbers into equation (1), we get






This is the desired value.