More precisely, you can think of the "1"s as being differently labelled at first, and then you see that exactly 6 configurations arise when you remove their labels.
I'm not asking you what to do. I'm asking you a question, and you must answer accordingly to understand.
The removal of labels on "1"s is a process that causes some configurations to become identical. For each final configuration, how many original configurations corresponded to it?
@Learner I saw your question. I'm not that interested in Stolz-Cesaro, since I never needed it to solve any limit/convergence question, so I am not familiar with it to give you an immediate answer. But have you tried to prove/disprove it yourself using the same proof as for the original theorem? If not, why not?
@user963241 There's not much meaning to "configuration". As I said just now, you permute all 8 objects, which gives you 8! permutations. I call each of these permutations a configuration, because we're going to deal with other configurations. After the label removal of "1"s, you get some arrangements, which I also call configurations.
Right, but you must stop thinking of them as "duplicates" or whatever you're currently thinking of.
Instead, you must focus on the fact that you have 6 original configurations corresponding to 1 final configuration (after the "1" label removal process).
Do you completely understand that?
@Learner You should include your attempted proof in your question, and add the tag proof-verification and other people are likely to answer it for you. If you get no answer, feel free to ping me here, but I may be busy in the next few weeks.
Not just fluent in Maths. Let's think of a simple example instead. Say, I have 100 pizzas and I give half of them to my friend and 90% of the remaining to another friend. So, I have: 100 pizzas and half of them is 50 pizzas. I give 90% of 50 i.e. 50 / 10 = 5 remaining pizzas. (100 / 2 * 10) = 5 pizzas.
@user963241 By saying "not just fluent in Maths" you are telling everyone that you are fluent in Maths and other things too. I think this clearly shows that you are not fluent in English. So just say "no".
And you're also not fluent in mathematics, yet.
I will repeat what I said in simpler terms. If you still cannot understand it, then find someone who can explain it to you.
@MohammadZuhairKhan What happened is not relevant to you, but basically I'm not interested in teaching people who waste my time. I have many other things I want to do.
The general form of Stolz-Cesaro $\infty/\infty$ case states that any two real two sequences $a_n$ and $b_n$, with the latter being monotone and unbounded, satisfy
$$\liminf\frac{a_{n+1}-a_n}{b_{n+1}-b_n}\le\liminf\frac{a_n}{b_n}\le\limsup \frac{a_n}{b_n}\le \limsup\frac{a_{n+1}-a_n}{b_{n+1}-b_n...
How to prove the $0/0$ case of Stolz-Cesaro Theorem? In other words:
Given that
$$\lim_{n \to \infty} a_n = \lim_{n \to \infty} b_n = 0$$
with $(b_n)_{n=1}^{\infty}$ strictly monotone, and that $$\lim_{n \to \infty} \frac{a_{n+1} - a_{n}}{b_{n+1} - b_{n}} = L$$ prove that $\lim_{n \to \infty} ...
Is a (non-straight) curve in the xy-plane considered to be 1-dimensional or 2-dimensional, and what is the term to used describe its other attribute (either its oneness, consisting in its similarity to a line, or its twoness, consisting in its need to be defined by at least two axes)?
How to prove the $0/0$ case of Stolz-Cesaro Theorem? In other words:
Given that
$$\lim_{n \to \infty} a_n = \lim_{n \to \infty} b_n = 0$$
with $(b_n)_{n=1}^{\infty}$ strictly monotone, and that $$\lim_{n \to \infty} \frac{a_{n+1} - a_{n}}{b_{n+1} - b_{n}} = L$$ prove that $\lim_{n \to \infty} ...
