A student John starts with the
familiar series 1 + 1/2 + 1/4 + 1/8 + . . . .

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?