Question 1109139
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The sum of an infinite geometric progression converges if and only if the common ratio is less than 1 by the modulus.  In your case it means

|2n -1| < 1,   or,  equivalently,


{{{abs(n-1/2)}}} < {{{1/2}}}.


The solution to the last inequality are all those "n" that are remoted less than {{{1/2}}}  from {{{1/2}}}, i.e.

    0 < n < 1.


Or, in the interval notation,  those "n" that belong to the interval (0,1)   (the ends are not included !).
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Solved.