How can I prove that   $$ \sum_{n=1}^{\infty}[ 2\, n\,\pi\coth^{-1}(2n)-\pi]= \frac{\pi}{2}-\frac{\pi\,\ln(2)}{2}$$  This result comes from using the Infinite product of $\sin(x)$ in the following well-known integral.   $$\int_0^{\frac{\pi}{2}} \ln[\sin(x)] $$  I hope to find a way to directly prove the result , rather than showing it is related to the above integral.  I have spent a great deal of time on this , so any hints or results will be greatly appreciated.   Thank you very much for your help and time.
how can I find sum of series $ \sum_{n=1}^{\infty} (n\cdot \ln \frac{2n+1}{2n-1} - 1) $? It is so weird for me because I put this to Mathematica and it tells me that sum does not converge...  Let consider sum no to infinity, but to n  $$ \sum_{k=1}^{n} (k\cdot \ln \frac{2k+1}{2k-1} - 1) =$$ $$ ln \frac{3}{1}\cdot \left(\frac{5}{3}\right)^2 \cdot...\cdot \left(\frac{2n+1}{2n-1}\right)^n - n = ln \frac{1}{1}\cdot \frac{1}{3}\cdot \frac{1}{5}\cdot ... \frac{1}{2n-1} - n $$ but $$ n = ln e^n $$ so it will be $$ln\frac{1}{e^n} \cdot \frac{1}{1}\cdot \frac{1}{3}\cdot \frac{1}{5}\cdot ... \frac{1}{2n-1}$$  So the limit of it is $-\infty$ Have I done this well or I missed sth?
Avoiding NullPointerException in Java I use object != null a lot to avoid . What is an alternative to: if (someobject != null) {     someobject.doCalc(); } Every time I open a folder from the desktop, a prompt is shown that the said folder is opened. How do I stop that? The use of "real" in the following cases Possible Duplicate:             If you listen real close...      Can you swing by real quick...   Sentences like the above two are what I often hear in daily life. If I didn't hear them in the real world, I would probably be more ready to say "If you listen really close..." and "Can you swing by really quick...".  Is real being used as an adjective in this scenario? If so, is the usage of adj.+ adj. a common practice in English language? Do I need a transit visa for a layover in the United Arab Emirates? What is $\int_{-\pi}^\pi \cos(nx)\cos(mx)\,dx$? I'm pretty sure that there's a theorem that says that the Fourier coefficients of a sum of $\cos(nx)$ and $\sin(nx)$ 's are the coefficients of the sum itself.   I tried to prove that in the specific case of $f(x) = \sum_{n=1}^{\infty}\frac{\cos(nx)}{n^2}$. In calculating the Fourier coefficients I got to calculating  $\cos(nx)\cdot\cos(mx)$, but I'm now stuck. Any tips would be appreciated! Fspy add-on not installing How to draw a torus Is there an easy way to draw a contour image of torus below with tikz? Or for that matter with any other graphics package. $P(\limsup A_n)=1 $ if $\forall A \in \mathfrak{F}$ s.t. $\sum_{n=1}^{\infty} P(A \cap A_n) = \infty$ Let $\mathfrak{F} = (A_n)_{n \in \mathbb{N}}$. Prove that $P(\limsup A_n)=1$ if $\forall A \in \mathfrak{F}$ s.t. $P(A) &gt; 0, \sum_{n=1}^{\infty} P(A \cap A_n) = \infty$. (Side question 1 Is second Borel-Cantelli out because we don't know if the $A_n$'s are independent?)  Suppose $\forall A \in \mathfrak{F}$ s.t. $P(A) &gt; 0, \sum_{n=1}^{\infty} P(A \cap A_n) = \infty$, but $P(\limsup A_n)&lt;1$. My prof gave us this convenient hint: Show $\exists M &gt; 0$ s.t. $P(\bigcap_{j=M}^{\infty} A_j^{c}) &gt; 0$.  $1 &gt; P(\limsup A_n)$  $1 - P(\limsup A_n) &gt; 0$  $P(\liminf A_n^{c}) &gt; 0$  By definition of $\liminf A_n^{c}$, $\exists M &gt; 0 $ s.t. $P(\bigcap_{j=M}^{\infty} A_j^{c}) &gt; 0$. (Side questions 2 and 3 Isn't this =1? ...Wait, is that what we're trying to prove? Haha)  $\to 1 - P(\bigcup_{j=M}^{\infty} A_j) &gt; 0$  $\to 1 &gt; P(\bigcup_{j=M}^{\infty} A_j)$  Main question: Now I guess we have to come up with some A s.t.  $\to 1 &gt; P(\bigcup_{j=M}^{\infty} A_j) \geq \sum_{n=1}^{\infty} P(A \cap A_n)$, or is there something else to do? Attribute field names truncation when creating shapefiles I am wondering what is the expression that truncate a given name to the 10 characters name used in ESRI shapefiles when an object is saved as shapefile with the function writeOGR in R. For example, the attribute field from a spatial dataframe "Ablepharus kitaibelii" is converted to "Ablphrk" when saved as shapefile, how the truncation is done? C: pointer to struct in the struct definition How can I have a pointer to the next struct in the definition of this struct:  typedef struct A {   int a;   int b;   A*  next; } A;   this is how I first wrote it but it does not work. PHP: running scheduled jobs (cron jobs) Let $R$ be a commutative ring with identity and $I$ and $J$ be ideals in $R$.     Show that $R/I\otimes_R R/J\cong R/(I+J)$ as $R$-modules.   This is what I tried.  Define a map $f:R/I\times R/J\to R/(I+J)$ as $f(\bar r, \bar s)=rs+(I+J)$. This is a well-defined bilinear map.  By the universal property of tensor products, we get an $R$-linear map $F:R/I\otimes_R R/J\to R/(I+J)$ which sends $\bar r\otimes \bar s$ to $rs+(I+J)$.  This map is surjective.  So all we need to show is that the kernel of $F$ is $0$.  But I am unable to do this. Why does an argument similiar to 0.999...=1 show 999...=-1? Extruding Vector Files in Blender I am sure this has been discussed quite a few times, but I found a tutorial recently doing the exact same thing as I did, but in my case my logo is messing up!  Here is the link to the workflow:   This is what it looks like after adding the solidify modifier:   I already recalced the normals of my spline, but besides that it messes up my spline shape, it had no effect.  Any suggestions on how to fix it? Tags in spoilers are always visible Discuss convergence of the series $\sum \frac{(1+n)^n}{n!}x^n$. Let    $$a_n = \dfrac{(1+n)^n}{n!}$$   Discuss convergence of the series $\displaystyle \sum a_nx^n$.   I've shown that the series converges absolutely when |x| &lt; 1/e and diverged when |x| > 1/e. I can't seem to determine what happens at |x| = 1/e A linking verb is a verb which connects a subject to its predicate   without expressing an action. A linking verb is used to re-identify or   describe its subject.      Ex:      Alan is a beast.      Alan seems drunk.   But I would say that some linking verbs can be used with "continuous tenses" &amp; other linking verbs can not, but I am not sure.      (linking verb) to experience a particular feeling or emotion      Ex: I’m feeling a little better today.      Do you still feel hungry?   So, feel in this meaning can be used with continuous tenses  But I would think this sentence "Do you still feel hungry?" does not make much sense because it uses "simple present tense" which expresses something that happens regularly. Noone would feel hungry regularly.  So, that sentence "you feel hungry" would mean "you always feel hungry".      (linking verb) (not used in the progressive tenses): to have a   particular physical quality which you become aware of by touching         adj.         The water feels warm.      Its skin feels really smooth.    says "Speakers sometimes use the Simple Present to express the idea that an action is happening or is not happening now. This can only be done with Non-Continuous Verbs and certain Mixed Verbs."  so, "The water feels warm." could mean "The water is always warm" (simple present expressing things that happen regularly - more or less permanent action)  But "The water feels warm." could mean "The water is warm now" (simple present with non-continuous verb expressing things that is happening now - a temporary action)     : (linking verb) to seem; to appear         adj.         to look pale/happy/tired      That book looks interesting.   In dictionary,      (linking verb) (not used in the progressive tenses): to   give the impression of being or doing something         adj.         She didn't appear surprised at all.       (to somebody) (to be) something (not used in the progressive tenses): to give the impression of being or doing something (synonym:   appear)         adj.         You seem happy.   So, we cannot say "you are seeming happy" or "you are appearing happy".  But I am not sure I can say "You are looking good" (sounds right to my ears) or "That book is looking interesting." (doesn't sound right to my ears)  But we can say "The future’s looking good." ()  So, some verbs has the same meaning but one can be used with "continuous" &amp; the other can not, right?  Ex: "you are looking happy", but "you seem happy". Difference between projection and datum? What's the difference between a projection and a datum? When installing user applications, where do "best practices" suggest they be located? Occasionally I install applications manually, rather than using apt or another package manager.   What location (/usr/, /usr/local/, /opt/, /home/, etc.) is suggested by "best practices" for the installation of user applications? I'm a US citizen and I've already gotten my Tier 4 visa to study in the UK in my passport, however, it looks like the start date on the visa and the letter to present to the Immigration Officer has allowed me entry an extra week on top of the usual month. My school starts on September 24th, but my visa states permission to enter on August 17th. I've read that you are only allowed a month before the beginning of school, but my visa is saying differently!  Should I assume its wrong and enter on the 24th or is it okay for me to enter on the 17th?   Thanks.