<s>[INST] Summation of $ \frac{1}{2} + \frac{3}{4} + \frac{7}{8} + \frac{15}{16} + \cdots$ till $n$ terms What is the pattern in the following?

  
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*Sum to $n$ terms of the series: $$ \frac{1}{2} + \frac{3}{4} + \frac{7}{8} + \frac{15}{16} + \cdots$$
  

 [/INST] So, what you want is basically $$\left(1 - \frac {1} {2}\right) + \left(1 - \frac {1} {4}\right)  + \cdots + \left(1 - \frac {1} {2^n}\right) = n - \left(\frac {1} {2} + \frac {1} {4} + \cdots + \frac {1} {2^n}\right) = n - 1 + \frac {1} {2^n}.$$
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