Hello, I need a help with the following:
1. Let $A$ be a transitive set, prove that $A\cup \{A \}$ is also transitive.
2. Show that for every natural $n$ there is a transitive set with $n$ elements.
Hello!
I've a problem understanding the following lines(Arzela lemma, and first two sentences of a proof) from Fichtengoltz's book.
I know, that some(2 members?) of you know Russian, help me please translate these line into English, with a short explanation on bold lines.
Thank you!
Let us define the following infinite sum:
$$\sum_{n=1}^{\infty} \frac{(\ln n)^n}{[\ln (n+1)]^{n+1}} $$
That sum is converges, use Cauchy root test, hence,
$$\lim_{n\to\infty}\frac{(\ln n)^n}{[\ln (n+1)]^{n+1}}=0$$
I'm little late to work, so just try to answer my question:
What is $$\sum\limits_{i=2}^n \frac{i}{x^i}$$ ?
$$\sum\limits_{i=2}^n \frac{i}{x^i}=\frac{2}{x^2}+\frac{3}{x^3}+\frac{4}{x^4}+...+\frac{n}{x^n}$$
Yes?
So, why then the above equals to what I wrote in my previous post? Good-luck! :)
After substitution $x=\frac{1}{t}$, you will get:
$$ \int_{0}^{\frac{\pi}{2}} \frac{\ln\cos (x)}{x^2}dx $$ Which can be written as:$$ \int_{0}^{\frac{\pi}{4}} \frac{\ln\cos (x)}{x^2}dx+ \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\ln\cos (x)}{x^2}dx $$ Could you proceed? (Hint: this integral is...
Let $f(x)$ be generating function of $T_n$:
$$ f(x)=T_0+T_1x+T_2x^2+...$$Hence, we have:
$$ f(x)=\sum_{i=0}^{\infty}T_ix^i=1+ \sum_{i=2}^{\infty}T_ix^i=1+\sum_{i=2}^{\infty}(aT_{i - 1} + bn)x^i $$
$$=1+\sum_{i=2}^{\infty}aT_{i - 1}x^i+ \sum_{i=2}^{\infty}bix^i= 1+a\sum_{i=2}^{\infty}T_{i -...