Concerning the sequence ∑p≤n,p⌊lognlogp⌋/n \sum_{p \le n , p} \big\lfloor\frac{\log n}{\log p}\bigr\rfloor/n where p is prime

The sequence (∑p≤n,pisprime⏟⌊lognlogp⌋n)∞n=1 is bounded , my question is what is the lim sup of this sequence ? Is the sequence convergent ? If it does , what is the limit ? Answer We have1n∑p≤n⌊log(n)log(p)⌋=1n∑p≤n|{m∈N+:pm≤n}|=1n∑pm≤n1=π(n)n+O(log(n)√n)=O(1log(n)+log(n)√n)=O(log(n)√n). where π(n)=∑p≤n1. Expanding a little bit, note that 1)|{m∈N+:pm≤x}|=|{m∈N+:mlog(p)≤log(x)}|=⌊log(x)log(p)⌋ 2)We have the bound |{p:pm≤x,m≠1}|=O(√xlog(x)) 3)By Prime Number Theorem π(x)=(xlog(x)). AttributionSource : Link … Read more

Definition of Strong Convergence in LpL^p

Is strong convergence in Lp, ie fistrongly⟶, just ||fi−f||Lp→0. If so why dont we just call it convergence? Answer Yes your definition is correct. Usually people do simply say convergence, but if you want to really emphasise that you do not mean another form of convergence e.g. weak convergence or almost everywhere convergence, you can … Read more

Convergent series and divergent series ∞∑n=1nn3+1\sum\limits_{n=1}^\infty \frac{n}{n^3+1} and ∞∑n=1n2+1n3+1\sum\limits_{n=1}^\infty \frac{n^2+1}{n^3+1}

Hi I have two questions. First, ∑∞n=1nn3+1. Is it divergent or convergent? I think it seems like it is positive and decreasing function so we can apply integral test. however, integrating this function is not an easy task. Is there any other test I can use? Second, ∑∞n=1n2+1n3+1. I can’t think of any test,, Can … Read more

How to show \sum_{n=1}^{\infty} (\sqrt{b_{n}}- \sqrt{b_{n+1}}) \sum_{n=1}^{\infty} (\sqrt{b_{n}}- \sqrt{b_{n+1}}) converges?

Let a_{n} \ge 0 \hspace{1cm} \forall n \in \mathbb{N} \cup \{0\}. and \sum_{n=1}^{\infty} a_{n} converges and b_{n}=\sum_{k=n}^{\infty} a_{k} Then we have to prove that \sum_{n=1}^{\infty} (\sqrt{b_{n}}- \sqrt{b_{n+1}}) converges. My try: b_{n+1} = \sum_{k=n+1}^{\infty} a_{k} =\sum_{k=1}^{\infty} a_{k} – \sum_{k=1}^{n} a_{k} \implies b_{n+1} = \sum_{k=1}^{\infty} a_{k}- \sum_{k=1}^{n-1} a_{k} -a_{n} \implies b_{n+1}= b_{n}-a_{n} And this gives, b_{n}- b_{n+1}=a_{n} … Read more

$\sum_{n=1}^\infty a_n<\infty$ if and only if $\sum_{n=1}^\infty \frac{a_n}{1+a_n}<\infty$

For $a_n$ positive sequence. I think I can prove one direction, but not both. Answer Since $\frac{a_n}{1+a_n}<a_n$, thus $\sum_{n=1}^\infty a_n<\infty$ implies $\sum_{n=1}^\infty \frac{a_n}{1+a_n}<\infty$. On the other hand, let $\sum_{n=1}^\infty \frac{a_n}{1+a_n}<\infty$. Then $\lim_{n\to\infty}\frac{a_n}{1+a_n}=0$ and hence $\lim_{n\to\infty}a_n=0$. So there is $N>0$ such that $a_n<1$ when $n\ge N$. From this, we have $$ \frac{1}{2}a_n\le\frac{a_n}{1+a_n}, \text{ for }n\ge N … Read more

Almost sure convergence of max\max(X_1, X_2,\ldots,X_n).

X_1, X_2,\ldots, X_n are independent with uniform distribution on [0,a]. Prove that \max(X_1, X_2,\ldots,X_n) \rightarrow a almost surely. Ar first I look for the probability distribution i.e. F_\max(t)=P(\max(X_1, X_2,\ldots,X_n)<t)=P(X_1<t)^n=(F_{X_1}(t))^n=\left(\frac{t}{a}\right)^n. So if t<a it converges to 0 and if t\geq a it converges to 1. How can I say that \max(X_1, X_2,\ldots,X_n) \rightarrow a almost surely. … Read more

Does fn(x)=cosn(x)(1−cosn(x))f_n(x)=\cos^n(x)(1-\cos^n(x)) converge uniformly for xx in [π/4 , π/2][π/4 , π/2]?

Does f_n(x)=\cos^n(x)(1-\cos^n(x)) converge uniformly for x in [π/4 , π/2]? Its clear to see that the point-wise convergence is to 0. By finding the derivative I obtained that the maximum of f_n is when \cos^n(x)=1/2 and that \sup |f_n(x)-0|= 1/2\cdot(1-1/2)=1/4 which would indicate that the function does not converge uniformly, however I’m not sure the … Read more

Is convergence in probability sometimes equivalent to almost sure convergence?

I was reading on sufficient and necessary conditions for the strong law of large numbers on this encyclopedia of math page, and I came across the following curious passage: The existence of such examples is not at all obvious at first sight. The reason is that even though, in general, convergence in probability is weaker … Read more

Uniqueness of a Limit epsilon divided by 2?

I have been reading about this theorem in a book called ‘Calculus: Basic Concepts for High-schools’, it is a very good book (so far) and I can highly recommend it. Well the author goes on to prove that: ‘A convergent sequence has only one limit’, which is quite intuitive in geometrical sense.. However in the … Read more

Task with convergence: ∫+∞0earctanxx2dx\int_{0}^{+\infty }\frac{e^{\arctan x}}{x^{2}}dx and ∫+∞0earctanx1+x2dx\int_{0}^{+\infty }\frac{e^{\arctan x}}{1+x^{2}}dx

a) \int_{0}^{+\infty }\frac{e^{\arctan x}}{x^{2}}dx b) \int_{0}^{+\infty }\frac{e^{\arctan x}}{1 + x^{2}}dx I know that is same , but I need to first check convergence which I think that when I put infinity e^{\frac{\pi}{2}} and then that get out and I have \int_{0}^{+\infty }\frac{1}{x^{2}}dx that convergent because p>1 And now I need to calculate it. And there … Read more