Does a $\limsup_{n\to\infty}$ exist for $\overline{\dfrac{n}{\phi (n)}},$ where $\phi$ is the Euler totient function, and the overbar represents the mean.  Of course $\dfrac{n}{\phi (n)}$ is unbounded, as it grows like $e^{\gamma } \log (\log (x))+\dfrac{\text{c}}{\log (\log (x))}$ for some $c\approx 3,$ where $c\rightarrow 0$ as $n\rightarrow \infty,$ but I was wondering whether the set of (mostly, priomorial multiples) is dense enough to impose a limit on the function.  Numerical tests suggest $\limsup_{n\to\infty}\overline{\dfrac{n}{\phi (n)}}\approx 2,$ but I can't find any references to the subject. Is this realistic, or is the function unbounded?
123407
There is a second part of the problem posted in , from Apostol's book, but I can't figure it out. It asks the reader to prove that if $x \geq 2$ then $\sum\limits_{n\leq x} \frac{n}{\phi(n)} = O(x)$, where the right hand side of the equation uses the Big-O notation. I've tried using the prior result, but I always end up with $O(x^2)$ . Any insights?
32091
We have $$\dfrac{1+2+3+...+ \space n}{n^2}$$  What is the limit of this function as $n \rightarrow \infty$?  My idea:  $$\dfrac{1+2+3+...+ \space n}{n^2} = \dfrac{1}{n^2} + \dfrac{2}{n^2} + ... + \dfrac{n}{n^2} = 0$$  Is this correct?
36532
Question:  Consider $E_n\in(0,1)$ and $E_n\to0,\text{as }n\to\infty$      Assuming that $E_{n+1} = E_n E_{n-1}$, show that, for a real $C$ independent of $n$, $$E_{n+1} \leq C{E_n}^\phi$$ where $\phi=\frac{1+\sqrt{5}}{2}$, the Golden Ratio   My attempt so far:  We can take logarithms of both side to see that   \begin{align*}\log(E_{n+1}) &amp;= \log(E_n E_{n-1})\\ &amp;=\log(E_n)+\log(E_{n-1})\end{align*}  We can see that this fulfils the conditions of the Fibonacci Sequence, $F_{n+1} = F_{n}+F_{n-1}$  Therefore, we can assume that $$\frac{\log(E_{n+1})}{\log(E_n)}\to \phi,\quad\text{as }n\to\infty$$  We can then say that \begin{align*}\lim_{n\to\infty}\left|\frac{\log(E_{n+1})}{\log(E_n)}-\phi\right|&amp;=0\\ \lim_{n\to\infty}\left|\log(E_{n+1})-\phi\log(E_n)\right|&amp;=0\\ \lim_{n\to\infty}\left|\log(E_{n+1}) -\log({E_n}^\phi)\right|&amp;=0\\ \lim_{n\to\infty}\left|E_{n+1} - {E_n}^\phi\right|&amp;=1\end{align*}  I'm not entirely sure that the last line of this is correct - I have attempted to get rid of the logarithms as I would have done in an equation with no limits or absolute values in. Could someone let me know if I have done anything wrong please. I am also struggling to continue from this point so any guidance would be much appreciated too.
2677
Here are two equivalent definitions of $\limsup_{n\rightarrow\infty} a_n$:   Let $u_n=\sup\{a_n, a_{n+1}, a_{n+2},\ldots\}$. Then $$\limsup_{n\rightarrow\infty} a_n = \lim_{n\rightarrow\infty} u_n =    \lim_{n\rightarrow\infty}\left(\sup\{a_n,a_{n+1},\ldots\}\right)$$ Let $E$ be the set of all subsequential limits of $\{a_n\}$. Then $$\limsup_{n\rightarrow\infty} a_n = \sup E$$   I'm curious as to which one people usually learn first, or which one people find more intuitive. Baby Rudin only has the second definition, but doesn't mention the first as far as I know.  As far as intuition, I think of the second definition as "the biggest limit point." I'm not sure how to think of the first definition though. I can see that $u_n \ge u_{n+1}$ for all $n$, but I'm not sure how to interpret the limit of those $u_n$ conceptually.  Are there any other ways to conceptualize $\limsup$?
116728
$\lim_{x \rightarrow \infty} \lim_{n \rightarrow \infty} f_{n}(x) = \lim_{n \rightarrow \infty}\lim_{x \rightarrow \infty}f_{n}(x) $ in case of uniform convergence of $f_{n}(x)$ to $f(x)$  I used this while I was given that $f_{n}(x)$ converges to $f(x)$ uniformly. Is this correct and I was also thinking to prove this!
20256
How can I compute the following limit? $$ \lim_{n\to\infty}\frac{\dfrac{n}{1}+\dfrac{n-1}{2}+\dots+\dfrac{2}{n-1}+\dfrac{1}{n}}{\ln(n!)} $$  I have tried lots of methods, I can't get the answer. Although I think the limit is $0$, I don't know how to explain it. Please, if someone could help me it would be fantastic.
70674
I am trying to show that for nonnegative $f$ on $\mathbb{R}$ if $||f||_1&lt;\infty$, we have  $$ \lim_{p\uparrow \infty}||f||_p=||f||_\infty $$ I have tried to fiddle around with  $$ \left(\int_\mathbb{R}|f|^p|f|^{1-p} \right)^{1/p}&lt;\infty $$ but haven't found a nice way to get an estimate on $\int_E|f|^p$ from this. Honestly I am pretty uncomfortable with non finite measure spaces.   Any hints would be awesome! Any references for problems similar to this would also be awesome.  edit: this is not a duplicate of the linked question as 1) my measure space is not finite, 2) my function is in $L^1$ and not necessarily in $L^\infty$
103504
I was wondering if there is a function $f: \mathbb{R} \to \mathbb{R}$ such that its limit at every point is infinite.   I guess not, because what would its graph look like, but then again, I don't know how to prove it.
20154
Could be the following limit computed without using Stirling's approximation formula?  $$\lim_{n\to\infty} \frac{1}{\sqrt[n+1]{(n+1)!} - \sqrt[n]{(n)!}}$$  I know that the limit is $e$, but I'm looking for some alternative ways that doesn't require to resort  to the use of Stirling's approximation. I really appreciate any support at this limit. Thanks.
20698
$$\displaystyle  \lim_{n \to \infty}{\dfrac{\Gamma\left(\frac{p}{m}+n+1\right) \Gamma\left(\frac{q}{m} + \frac{n}{2} + 1\right)}{\Gamma\left(\frac{q}{m}+n+1\right) \Gamma\left(\frac{p}{m} + \frac{n}{2} + 1\right)}}$$  This is from .   If we add one more variable -      $$\displaystyle L(p,q,m,t) = \lim_{n \to \infty}{\dfrac{\Gamma\left(\frac{p}{m}+n+1\right) \Gamma\left(\frac{q}{m} + \frac{n}{t} + 1\right)}{\Gamma\left(\frac{q}{m}+n+1\right) \Gamma\left(\frac{p}{m} + \frac{n}{t} + 1\right)}}$$   Then, the results I got from W|A are really interesting. They seem to follow the pattern which says      $\displaystyle L(p,q,m,t) = \dfrac{1}{t^{|p-q|/m}}$   If it is indeed true, how to prove it? Factorial approximations did not lead me anywhere.
119725
As the title states, I'm interested in the asymptotic behavior of $$\sum\limits_{k=1}^n \frac{1}{k^{\alpha}} , \alpha &gt; \frac{1}{2}$$ for $n \to \infty $. Any hints/ideas?
37682
Today my lecturer put up on the board that:  If $\lim\limits_{n\to\infty}\frac{a_{n+1}}{a_n}$ exists and $a_n&gt;0$ then   $\displaystyle \limsup\limits_{n\to\infty}\left(a_n^{\frac{1}{n}}\right)=\lim_{n\to\infty}\frac{a_{n+1}}{a_n}$  however I am not sure why this is true, can somebody give me a hint or something as to how to go about proving this.  thanks for any help
32639
How do take this limit:  $$ \lim_{n\to\infty} \frac{\sqrt{n}}{\log(n)}$$  I have a feeling that it is infinity, but I'm not sure how to prove it. Should I use L'Hopitals Rule?
83864
In fact the real question is: Does $\lim\limits_{n\to\infty}1^{n}=e$?.  I know that   $$ \lim\limits_{n\to\infty}\left(1+\dfrac{1}{n}\right)^n=e, $$  So, can we say that $1^\infty=e$?  And, by logic, This product  $$\underset{\text{infinity times}}{1\cdot1\cdot\ldots\cdot1},$$ gives $1$ not other value.
49586
Assume  $$\lim_{n\to \infty}  (y_{2n} - y_n)=0$$  then  $$\lim_{n\to \infty} y_n$$  exists.  I know it's not true, and I can see a sequence that disprove that $(1,1,2,1,3,2,4,1,5,3,...)$ but I want a sequence that has formula.  hint/help?  thanks
106773
Does $\sum\limits_{n=3}^{\infty} \dfrac{1}{(\log\log n)^{\log n}}$ converge or diverge ? I tried some tests , but nothing conclusive is coming . Pleas help
29696
I'm trying to solve this previous qual problem from my univeristy: Let $L_n$ be the continuous linear functions on $L^\infty(\mathbb{R})$ given by $$L_n(\phi)=\frac{1}{n!}\int_0^\infty x^ne^{-x}\phi(x)\,dx.$$  Prove that $L_n$ has no subsequence that converges in the weak* topology of $L^\infty(\mathbb{R})^\ast$. So far, I managed to see that $\frac{1}{n!}x^ne^{-x}$ converges uniformly on compact sets to 0, hence the weak* limit of any subsequence of $L_n$ must take compactly supported $L^\infty$ functions to 0. I've also tried to construct a test function directly for any subsequence but so far I have yet to come up with such a construction. Can anyone pleas help me? Some hints would be very much appreciated! Thanks!