>I agree. But the infinite sum, whether or not defined as the limit, *is* not a limit. A sum is the addition of summands --- and that is rational for every summand added. The adding does never stop, and the rational character of the result does never stop.
The sum is whatever we define it as, and infinite sum is defined as the limit of increasing finite sums.
>Of course, an infinite number of terms is more than any finite number. But that should not allow you to drop logic.
No one has dropped logic anywhere, this is an empty statement of yours.
>The word "infinite" seems to paralyse the common sense of many mathematicians.
Or more likely, you are too stupid to udnerstand it.
>If all terms fail and if there are infinitely many, then infinitely many fail.
Show an example of this using definitions.
>Why don't you comprehend this simple case?: If all terms fail and if there are infinitely many, then infinitely many fail.
Provide an example of this where an infinite sum, from definition, fails.
Remember that infinite sums are only defined for sequences that are cauchy.