It is well known that if you add together all of the unit fractions, 1 + 1/2 + 1/3 + 1/4 + 1/5 + … all the way to infinity, the answer is also infinity (although it approaches it ridiculously slowly).

However, if we throw out any that have a never-ending decimal, such as 1/3 (0.333…), 1/6 (0.166…) ,1/7 (0.142857…) etc, and only include those that have a terminating decimal expansion:

S = 1 + 1/2 (0.5) + 1/4 (0.25) + 1/5 (0.2) +1/8 (0.125) + 1/10 (0.1)… all the way to infinity,

we do get an actual number as the result. What is it?