The sum clearly diverges. This is indisputable. The point of the claim above, that
is to demonstrate that a sum of infinitely many terms can be manipulated in a variety of ways to end up with a contradictory result. It's an artifact of trying to do computations with an infinite number of terms.
The mathematician Srinivasa Ramanujan famously demonstrated the above as follows: Suppose the series converges to some constant, call it
. Then
Now, recall the geometric power series
which holds for any
. It has derivative
Taking
, we end up with
and so
But as mentioned above, neither power series converges unless. What Ramanujan did was to consider the sum 
as a limit of the power series evaluated at :
then arrived at the conclusion that
.
But again, let's emphasize that this result is patently wrong, and only serves to demonstrate that one can't manipulate a sum of infinitely many terms like one would a sum of a finite number of terms.
is to demonstrate that a sum of infinitely many terms can be manipulated in a variety of ways to end up with a contradictory result. It's an artifact of trying to do computations with an infinite number of terms.
The mathematician Srinivasa Ramanujan famously demonstrated the above as follows: Suppose the series converges to some constant, call it
Now, recall the geometric power series
which holds for any
Taking
and so
But as mentioned above, neither power series converges unless
then arrived at the conclusion that
But again, let's emphasize that this result is patently wrong, and only serves to demonstrate that one can't manipulate a sum of infinitely many terms like one would a sum of a finite number of terms.