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?
Get rid of the computer! Dumb brute force hides what is actually going on.
The actual series doesn't matter, just the first term and sum of the original series matters.
So first some investigation. Let our series be a + b + c + d + ...
Then applying the transformation creates a new series a/2 + (a+b)/4 + b/2 + (b+c)/4 + c/2 + (c+d)/4 + d/4 + ...
Apply a bit of simplification to get the equivalent sum 3a/4 + b + c + d + ...
This sum is a/4 less than the earlier series sum.
The sequence of first terms of the progressive series is a, a/2, a/4, a/8, ...
One quarter of each of these terms is deducted from the original sum S.
This evaluates to (1/4)*(a + a/2 + a/4 + a/8 + ...) = a/2
So for an original series whose first term is a and sum is S the limit of the process will result in a sum equal to S - a/2.
For the series present in the problem a=1 and S=2, then the limit for the original problem is 2 - 1/2 = 3/2.
Analytic Solution
