Find $\lim_{n\to \infty} \sum_{k=1}^n \frac{k}{n^2+k^2}$ Find $\lim_{n\to \infty} \sum_{k=1}^n \frac{k}{n^2+k^2}$   Since $\frac{k}{n^2+k^2}\leq \frac{k}{k^2+k^2}=\frac{1}{2k}$, then $\sum_{k=1}^n \frac{k}{n^2+k^2}\leq \sum_{k=1}^n \frac{1}{2k}=\frac{1}{2} \sum_{k=1}^n\frac{1}{k}$.  Now we send $n$ to infinity, then since $\sum_{k=1}^\infty \frac{1}{k}$ is harmonic, $\lim_{n\to \infty} \sum_{k=1}^n \frac{k}{n^2+k^2}$ doesn't exist.  I wonder if my thinking is right.
How do you calculate this limit $\lim_{n\to\infty}\sum_{k=1}^{n} \frac{k}{n^2+k^2}$? How to find the value of $\lim_{n\to\infty}S(n)$, where $S(n)$ is given by $$S(n)=\displaystyle\sum_{k=1}^{n} \dfrac{k}{n^2+k^2}$$   is unable to calculate it.  This is a question from a questions booklet, and the options for the answer are--  $\begin{align} &amp;A) \dfrac{\pi}{2} \\ &amp;B) \log 2 \\ &amp;C) \dfrac{\pi}{4} \\ &amp;D) \dfrac{1}{2} \log 2 \end{align}$
I looked up a few grammar-teaching websites (in Chinese language). The website says many people do not use the phrase "A xxx of" correctly because they do not identify the subject of the phrase correctly.   The website says phrases like "A range of; A high percentage of" is singular, while "a number of; a group of; a couple of; a total of" is plural.  Therefore according to the website, the correct sentences should be as follows.     A wide range of courses is available. ("A wide range" is the subject)      A high percentage of peptides that are made of amino acids is present   in the sample. ("A high percentage" is the subject)      A number/ group/ couple of people were killed in the earthquake.("people" is the subject )       A total of 230 employees were laid off. ("employees" is the subject)   I don't get the pattern here. All of this looks (or should I use "look"?) arbitrary to me. In fact I thought this phrase is supposed to be arbitrary and whether the phrase is plural or singular should depend on the the person's own judgment.   How do I identify the correct subject in the "A xxx of" phrase? Expansion of $(1+px+qx^2)^8\equiv 1+8x+52x^2+kx^3$... $(1+px+qx^2)^8 \equiv  1+8x+52x^2+kx^3...$  Given that expression, calculate the values of p, q, and k.  I'm guessing this is a trinomial that needs to be turned into a binomial, but how would you do it? I have no idea. How do I replace this cast iron toilet flange? iOS app crashes after deleting a comment NIST randomness test p values I have a binary sequence of length $2^{20}$. I am using the NIST statistical test, assess.   What should be taken for ./assess and  How many bitstreams? in this test? In the final Analysis report, what p-values should be there to pass this test? Introductory biology text for an outsider I'm a maths major and I have an interest in learning biology. I know very, very little; I know how babies are made and that's about it. Could anyone recommend a stimulating text to read for its own sake and also to use to learn biology? Understanding the proof for $\gcd (F_n, F_{n+k}) = 1$ where $F_n$ is a Fermat number I'm trying to understand the following proof of why      No two Fermat numbers have a common divisor greater than $1$.   Hardy and Wright show the following proofs:   If $x = 2^{2^n}$, we have  $$\frac{F_{n+k} - 2}{F_n}(\circ) = \frac{2^{2^{n+k}}-1}{2^{2^n}+1} = \frac{x^{2^k}-1}{x+1} = x^{2^k-1} - x^{2^k-2} + \ldots -1$$  The main issue I'm having is the $\circ$ part. I don't understand how they get that. The rest of the proof makes sense but the first part is really confusing me. Can someone please explain where they get that fraction?  Thanks a bunch! Prove that $\phi(n) \geq \sqrt{n}/2$ Solve for $A,B$: $\mathrm{LCM}(A,B)=168$, $\mathrm{HCF(A,B)}=12$ Sometimes when adding an object to the scene I see that all options (on toolbar or in window after pressing F6) are disabled. Can anyone tell me when can this happen? What is causing it and how to revert it - make it enabled again. It worked for me when I restarted Blender and reloaded the scene. However I would really like to know what can cause it? Converting from DMS to DD using Python in Field Calculator? I need to convert lat/long that is expressed as degrees, minutes, and seconds in the data into decimal degrees. For example, in the data, they are listed as N335042.06 in the Latitude column, and W86031.04 in the Longitude column. I have done this problem before where I created a script which converted DMS to DD, and vice-versa, so I guess I could use bits from that. But the problem I'm having is how to ignore (for lack of a better word) the 'N' and 'W' in the data? Can I skip them? And DMS are listed all together without any symbols or spaces.  Can I use len(), range(), split() to specify what part to read from the value? For example, can do the following?  N335042.06 where, 33 = degrees 50 = minutes 42.06 = seconds ...?  I came across  ESRI article, but it's in VB. Will probably use it as a reference but some of the terminology/syntax is different from Python.  Final code that works!   # Pre-logic def latDD(x):   D = int(x[1:3])   M = int(x[3:5])   S = float(x[5:])   DD = D + float(M)/60 + float(S)/3600   return DD  # Expression latDD(!Latitude!) Which media player has seek preview with thumbnails? I was looking for a good video player to replace VLC, something that is as good as vlc but has seek preview. I am tired of fast forwarding in vlc  EDIT: I meant seek preview with thumbnails Wireframe iOS design apps for iPad? Is there an app for iPad to assist in wireframe design for iPad and iPhone apps?  I don't mean actual development, just placing standard widgets and the things you usually do with photoshop and associated kits. How can I check if a word is contained in another string using PHP? How do I remove an OpenID or login? How do I remove login credentials (emails or OpenIDs) from my account?     For more information, see "" in the . A book suggests that we should not interchange "so" and "so that". "So" means "therefore", and "so that" means "in order that". However, it seems to me that in many cases they don't actually have a difference. For example:     Alice got up at 5:00 A.M., so that she could drive her son to school.   If I substitute "so that" with "so", this sentence presents the exact same causal relation to me:     Alice got up at 5:00 A.M., so she could drive her son to school.   I must have missed something here, since English is my second language. Could somebody explain the difference? What does \s! do? 1st example:  \starttypescript [\s!serif] [paratype]   \definefontsynonym [ParatypeSerif-Regular] [\s!name:PTSerif-Regular] [\s!features=\s!default]   \definefontsynonym [ParatypeSerif-Italic] [\s!name:PTSerif-Italic] [\s!features=\s!default]   \definefontsynonym [ParatypeSerif-Bold] [\s!name:PTSerif-Bold] [\s!features=\s!default]   \definefontsynonym [ParatypeSerif-BoldItalic] [\s!name:PTSerif-BoldItalic] [\s!features=\s!default] \stoptypescript   2nd example:  \definefont[SerifS][\s!file:GenBkBasR at \smallfontsize]   Is \s! only related to fonts? Should I use it in my typescript files? How to search the internet for terms with special characters I want to compile my program with the latest version of gcc.  Ubuntu 14.04 comes with gcc 4.8.2, however there's 4.9.0 available, moreover, I see that it is available as a package: . I tried to install it  sudo apt-get install gcc-4.9   but it says  Reading package lists... Done Building dependency tree        Reading state information... Done Note, selecting 'gcc-4.9-base' for regex 'gcc-4.9' gcc-4.9-base is already the newest version. 0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded.   Looks like it is already installed, just not as the default one? How do I utilize it to build my program? Can I transfer Windows 10 Pro license to another computer?