I was watching through professor dave's video https://youtu.be/L-JqHo4-W4k?list=PLybg94GvOJ9FoGQeUMFZ4SWZsr30jlUYK&t=433 and at 7:13 he claims that in order for a series to be convergent, its sequence must go to zero in the limit of infinity.
I can't entirely agree with this claim as I think it's fairly trivial to produce a series that converges while having it's sequence have a nonzero limit in infinity.
The example I am giving is below,
f(x) = 1/2^x
g(x) = f(infinite-x) (where infinite is the cardinality of N)
g(x) must converge as it has the same set of values as f. But the sequence at g(infinite) =
f(infinite-infinite) = f(0) = 1, does not go to zero.
I am trying to say like what if you flipped a converging function around horizontally, then it wouldn't go to zero at infinity, but the series on that function would converge.
Have I missed something about the nature of infinity? Is this already known and this video is a simplification of the full theorem?
Thank you, friends.
edit: I think my maths communication may not be the best so here are some examples
note: the sum of this sequence (the series) is 2.
I understand that potentially this may not be the most correct way of describing a function like this, but I think it does demonstrate my point?
The ratio test seems to work better for me but I would still like to know where my explanation is wrong.
The Divergence Test/Theorem from Calculus states that if $\sum_{n=0}^\infty a_n$ converges, then $\lim_{n\to \infty} a_n = 0$, as proved in Paul's Online Notes: https://tutorial.math.lamar.edu/classes/calcii/convergenceofseries.aspx
Edit: The problem presented in your example is in the ellipses. There is no next element if you reverse the ordering of the naturals. If my index set is $\{0\} \cup \{ 1/n : n \geq 1\}$, for example, after $0$, there is no next element. Moreover, arithmetic with infinity is not really tenable. For example, if you want to think of infinity as a limit, then we could maybe say $\infty = \lim_{n\to\infty} n$. Maybe then (loosely speaking) $\infty - \infty = \lim_{n\to\infty} n - \lim_{n\to\infty} n = \lim_{n\to\infty} 0 = 0$. However, $\infty = \lim_{n\to\infty} n + 1$ so then $\infty - \infty = \lim_{n\to\infty} n+1 - \lim_{n\to\infty} n = \lim_{n\to\infty} 1 = 1$, but clearly $0 \neq 1$.
Another obstacle to this reordering scheme is the following: series that converge conditionally can be rearranged and obtain different values in their sum. Though that's not a problem in your example since that series converges absolutely.