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Aug
20
comment $\int_{0}^{\infty}f\left(\frac{a}{x}+\frac{x}{a}\right)\frac{\ln x}{x}dx=\ln a\int_{0}^{\infty}f\left(\frac{a}{x}+\frac{x}{a}\right)\frac{dx}{x}$
+1 for a slick solution - for clarity's sake, the top integral here is the difference between the two terms in the OP's equation (using $\log x-\log a=\log\frac xa$).
Aug
19
comment What's the order of growth of the 'double-and-rearrange' numbers?
@r.e.s. Yes, but the question asks (for convenience) about $\leq n$-digit integers (since the total number of these is $10^n$, possibly $-1$ if you don't count zero), so the total count of those numbers with all digits odd is $\sum_{i\leq n}5^i$.
Aug
19
comment Is it possible to construct a sequence that ends in 1000000000?
I got curious enough to ask my order-of-growth question: math.stackexchange.com/questions/1403089
Aug
19
asked What's the order of growth of the 'double-and-rearrange' numbers?
Aug
19
comment Finding closed form of finite product
I suggest posting up the precise problem you've been given and the work you've done on it so far; that's much more likely to get you a useful answer than trying to work things out from the little subproblem you've selected.
Aug
19
comment Is it possible to construct a sequence that ends in 1000000000?
So many interesting questions here - I'm not even sure of the rough order of growth of $f(n)=\#\{x\leq n:$ there exists a double-and-rearrange sequence reaching $x\}$...
Aug
19
comment Is it true that Ackermann's function cannot be implemented without recursion?
You might want to have a look at math.stackexchange.com/questions/96483/… , whose top answer gives a pretty good characterization of 'primitive recursive' that might offer an answer to your question.
Aug
18
comment Finding closed form of finite product
Is your task to find the closed form, or is it to solve some other problem? There are certainly closed forms for the expression listed, but none of them are really likely to come from the given hint...
Aug
18
comment Simple question, what is meant by 'as $x \to \infty$ the number of squares $\leq x$ is $\sqrt{x} + O(1)$?
In this case, it means that if $sq(x)$ is the number of squares $n^2 \leq x$, then there's some constant $C$ and a $x_0$ such that $|sq(x)-\sqrt{x}|\leq C$ (i.e. $C\cdot 1(x)$) for all $x\geq x_0$.
Aug
18
comment Simple question, what is meant by 'as $x \to \infty$ the number of squares $\leq x$ is $\sqrt{x} + O(1)$?
What you know isn't quite so; for instance, the function $f(n)=(-1)^n$ is $O(1)$ even though the limit your definition talks about doesn't exist.
Aug
18
revised Prove that $\pi$ is a transcendental number
added 51 characters in body
Aug
18
comment $\liminf_{n\to\infty} \frac{\varphi(n)}{n} = 1$, not $0$
Most relevantly, changing any finite number of values of a function won't change its lim inf. And so in particular, examining any finite number of values can't possibly tell you the lim inf.
Aug
16
comment How to prove $3^\pi>\pi^3$ using algebra or geometry?
What's your geometric definition of $3^\pi$?
Aug
15
comment How many triangles exist whose angles are rational and side lengths are roots to quadratic equations?
Note that while wendy kriger's answer covers your specific question (values of the form $r+s\sqrt{n}$ with $r$ and $s$ rational and $n$ integer), there are other rational angles whose sines and cosines can be expressed using nestded square roots whose 'leaf values' are rational numbers (or equivalently, which are constructible with ruler and compass).
Aug
15
revised In the history of mathematics, has there ever been a mistake?
Added a bit of new news!
Aug
15
comment Finding a Recurrence Relation.
@Sudhanshu That looks right to me. But it's also true that $A_i(n)=B_i(n)$ for all $i$ : swapping all instances of $A$ and $B$ in a string in one class converts it to a string in the other class.
Aug
14
comment Finding a Recurrence Relation.
@Sudhanshu You won't get a direct recurrence relation for $S()$; rather, you'll get (entwined) recurrence relations for $A$ and $B$ - but that's not too complicated, since in fact there's a very obvious relation (what?) between $A_i(n)$ and $B_i(n)$.
Aug
14
comment Finding a Recurrence Relation.
@Sudhanshu Ahhh, I see. You can certainly build the recurrence relation, but you may be able to argue by symmetry and get an answer very quickly.
Aug
14
comment Finding a Recurrence Relation.
One curiosity question: does the problem say 'find a recurrence for $S(n)$' or does it ask you to find a formula for $S(n)$?
Aug
14
comment Finding a Recurrence Relation.
@Sudhanshu What do you mean by 'too large'?