One of my daughters was having a small programming exercise. 

Let's consider following algorithm:


Take a list of length $n$: $\ (1\,\ 2\,\ \ldots\,\ n)$.
Remove every $2$nd number.
From the resulting list, remove every $3$rd number.
From the resulting list, remove every $4$th number.
... Follow on until the list remains unchanged and let $u_n$ be the number of remaining elements.


Example with $n=11$


$(\ 1\,\ 2\,\ 3\,\ 4\,\ 5\,\ 6\,\ 7\,\ 8\,\ 9\,\ 10\,\ 11\ )\quad \Rightarrow\quad (\ 1\ *\ 3\ *\ 5\ *\ 7\ *\ 9 \ *\ 11\ )$
$(\ 1\,\ 3\,\ 5\,\ 7\,\ 9\,\ 11\ )\quad \Rightarrow\quad (\ 1\,\ 3\ *\ 7\,\ 9\ *\ )$
$(\ 1\, 3\,\ 7\,\ 9\ )\quad \Rightarrow\quad (\ 1\,\ 3\,\ 7\ *\ )$
$(\ 1\,\ 3\,\ 7\ )\ $ -- will not be modified anymore, and therefore $u_n=3$.


QUESTION: &nbsp; why do we have $\lim\limits_{n \to +\infty} \frac{n}{u_n^2}=\frac{\pi}{4}$ ?

Thanks!


Question: Making new verbs from the given words
Answer: clear -> clarify, meaning to make clear.

noble -> ennoble, meaning to make noble. That word is used in the sense of "make this person a member of the nobility" and also "make an action grand or self-sacrificing".

body -> embody, to put something into concrete form.

rate is already a verb. I guess you could say rate -> rate.

Likewise store is already a verb.