He then takes the average of each adjacent pair of terms and inserts it between the terms to obtain the new series 1 + 3/4 + 1/2 + 3/8 + 1/4 + . . . .
He divides this by two, because there are now twice as many terms as before. That gives
1/2 + 3/8 + 1/4 + 3/16 + 1/8 + ...
He repeats the process indefinitely. For example, the next pair of steps gives
1/2 + 7/16 + 3/8 + 5/16 + 1/4 + 7/32 + 3/16 + 5/32 + 1/8 + ...
then
1/4 + 7/32 + 3/16 + 5/32 + 1/8 + 7/64 + 3/32 + 5/64 + 1/16 + ...
What exact limit will the series approach?
