@Akbar2thegreat said in #36:
> #1
> Infinite set of elements is never empty. Element can be anything like 0, 1, 2.
0 + 0 + ... is not a set. It's an infinite series:
0 + 0 + ... = ∑_(n=1)^∞ 0
One could also write:
∑_(n=1)^∞ a_n,
where a_n = 0 ∀n ∈ Z+
> A set is empty when count of elements present in it is 0.
True, but irrelevant in this context.
> That summation (0+0+...), that tends to one!
Not at all. As others in this thread (m011235 and marcin2) have already pointed out the value of the titular infinite series is simply 0.
0 + 0 + ... = ∑_(n=1)^∞ 0 = 0
The above infinite series is convergent, since the sequence of its partial sums S_N converges. The N-th partial sum (N ∈ Z+, where Z+ = {1, 2, 3, 4, 5, ...} denotes the set of all positive integers) of our infinite series is defined as follows:
S_N = ∑_(n=1)^N 0
It is trivially easy to see that ∀N ∈ Z+ ("for all N element of the positive integers") the following holds: S_N = 0
(zero added to zero finitely many times, in this case N-1 times, will always yields zero).
So all partial sums are zero:
S_1 = 0
S_2 = 0 + 0 = 0
S_3 = 0 + 0 + 0 = 0
S_4 = 0 + 0 + 0 + 0 = 0
...
The partial sums S_N of an infinite series are ordered (which is important in the general case) and there are countably infinitely many of them. They thus form an infinite sequence (S_N)_N∈Z+. If said sequence converges, then the value it converges to is unique and is called the limit of the sequence:
lim_(N –> ∞) S_N
If the series of partial sums (S_N)_N∈Z+ converges, then the corresponding infinite series ∑_(n=1)^∞ 0 is said to be convergent and has a value equal to the limit of the sequence of its partial sums lim_(N –> ∞) S_N.
Because all S_N are zero |S_N - 0| = 0 ∀N ∈ Z+ and thus you can always (trivially) find an arbitrarily small (but positive) ε such that |S_N - 0| < ε for all N ≥ K, some finite positive integer. In fact, in this special case K can even be equal to 1. This constitutes convergence.
Thus the limit of (S_N)_N∈Z+ is simply equal to zero:
lim_(N –> ∞) S_N = 0
And finally the value of the corresponding infinite series is equal to zero:
0 + 0 + 0 + ... = ∑_(n=1)^∞ 0 = lim_(N –> ∞) S_N = 0
