Quote:
Originally Posted by RoboDesigners
I thought that because the series doesn't converge, it doesn't equal zero to begin with.  |
The original infinite sum you wrote:
(1 - 1) + (1 - 1) + (1 - 1)...
...
does converge. It converges because of the parentheses. The term that is being repeatedly added is (1-1). It
is equal to zero.
If you remove all the parentheses, so that you have
1 - 1 + 1 - 1 + 1 - 1 ...
then you are alternately adding plus or minus 1, so the sum never converges: it oscillates between 1 and 0.
The error was re-arranging the parentheses. The associative law does not always hold for an infinite sum: You cannot re-arrange the parentheses in an infinite sum unless certain criteria are satisfied.