What is the value of the sum $2^{-1} + 2^{-2} + 2^{-3} + \cdots + 2^{-9} + 2^{-10}$? Give your answer as a simple fraction.

The sum, S, is given by

S = (1/2) [ 1 - r^n ] [ 1 - r ]      where (1/2)  is the first term   [  1/2 =  2^1 ]      and r is the common ratio between terms = 1/2                                                 and n is the number of terms = 10

So we have

S = (1/2) [ 1 - r^10 ] / [1/2]   =

 [ 1 - (1/2)^10 ]  =

[ 2^10 - 1 ] / 2^10  = 

1023 / 1024

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