If $\lambda_1, \cdots, \lambda_n, \cdots$ are positive real numbers such that $\displaystyle\sum_{k=1}^{\infty}(\lambda_1+\cdots+\lambda_k)=+\infty$, we can affirm that $\displaystyle\sum_{k=1}^{\infty}\frac{\lambda_k}{(\lambda_1+\cdots+\lambda_k)^2}$ converges?  I try use some comparison test, but I have not success. Can someone help me? Thanks.
Assume that $0 &lt; a_n \leq 1$ and that $\sum a_n=\infty$. Is it true that  $$ \sum_{n \geq 1} \frac{a_n}{(a_1+\ldots+a_n)^2} &lt; \infty $$ ?  I think it is but I can't prove it. Of course if $a_n \geq \varepsilon$ for some $\varepsilon &gt; 0$ this is obvious.  Any idea? Thanks.
Strange error defining a macro with arguments for math mode What are the taxation rules for US non-resident aliens with respect to qualified dividends? Are the rules different for qualified versus non-qualified dividends? Assuming no DTAA exists between the US and the country of residence. Word for someone who acts like an expert but who has very little knowledge? What is a word to describe an individual who acts like an expert in a subject area, constantly stating facts and correcting people, but who actually has very little knowledge on the subject? Could not get lock /var/lib/apt/lists/lock when I try to apt-get update I'm getting the below error,  E: Could not get lock /var/lib/apt/lists/lock - open (11: Resource temporarily unavailable) E: Unable to lock directory /var/lib/apt/lists/ Design a custom Beamer theme from scratch Handling very large numbers in Python Change border color of \fbox Possible Duplicate:          I'm using an \fbox to have a border around a minipage.  \fbox{ \begin{minipage}{0.25\textwidth} my content \end{minipage} }   Is there any way to set the border color of the \fbox? How does long-distance space travel work in the MCU? In Guardians of the Galaxy vol. 2, we see long-distance space travel. It involves piloting your ship into some sort of "jump point" that opens up in space, taking the shape of a honeycomb-like structure with a single hex missing.  The way it's describe, these points are fixed in space: early in the movie, the Guardians have to out-run an enemy fleet to reach the one they need; later on, we see them fly through a very long series of them in sequence. It's also made clear that doing too many of these "jumps" in a row has consequences for the passengers.  The process seems to be similar to, say, the Mass Effect relay setup, mixed with a bit of Star Wars hyperspace travel.  Is this the first time we have seen long-distance space travel in the MCU? Is there anything from the source material that explains how this works from a logistical standpoint? If there's any information on the "science" behind it, that would also be helpful. How to remove cookies using CookieManager for a specific domain? I want to writing the matrix on the arrow from X to Y 3x3 type, from Z to Y 3x2 type ; I'm writing code to do Xml serialization. With below function.  public static string SerializeToXml(object obj) {     XmlSerializer serializer = new XmlSerializer(obj.GetType());     using (StringWriter writer = new StringWriter())     {         serializer.Serialize(writer, obj);         return writer.ToString();     } }   If the argument is a instance of class without parameterless constructor, it will throw a exception.     Unhandled Exception:   System.InvalidOperationException:   CSharpConsole.Foo cannot be serialized   because it does not have a   parameterless constructor.    at   System.Xml.Serialization.TypeDesc.CheckSupported()   at   System.Xml.Serialization.TypeScope.GetTypeDesc(Type   type, MemberInfo sourc e, Boolean   directReference, Boolean throwOnError)   at   System.Xml.Serialization.ModelScope.GetTypeModel(Type   type, Boolean direct Reference)    at   System.Xml.Serialization.XmlReflectionImporter.ImportTypeMapping(Type   type , XmlRootAttribute root, String   defaultNamespace)    at   System.Xml.Serialization.XmlSerializer..ctor(Type   type, String defaultName space)    at   System.Xml.Serialization.XmlSerializer..ctor(Type   type)   Why must there be a parameterless constructor in order to allow xml serialization to succeed?  EDIT: thanks for cfeduke's answer. The parameterless constructor can be private or internal. What is causing disabled options when adding a mesh/curve object to the scene? 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? This movie came out around 2000 and was created for one of the cable networks (and never in the theaters).  I think it was on the original Sci-Fi Channel or maybe the USA Network, but really not sure.    It's about a 1950's American-style utopian world.  The main character, if I remember correctly, worked at a car dealership.  I recall him looking like Jon Hamm, Adrian Pasdar, or Scott Bakula, but I may be way off there (and I didn't see the movie in their filmographies).  He drives his nice car home, and the whole town is beautiful, as are its people and white picket fence style homes.     He suddenly starts seeing weird things.  For example, instead of people driving cars, they're actually riding rickety bicycles.  The manicured, paved streets are actually dirt roads.  Houses are, in reality, tiny shacks.  Instead of people's clothes being immaculately maintained, everyone's really wearing filthy rags.  And instead of delicious food, they're eating something that looks like rotten garbage.  Something is making everyone believe they live in a beautiful, perfect society, but it's mind manipulation that's almost on the scale of the movie "The Matrix."    He befriends a pretty lady, and together they travel outside their town and try to find answers.  They even meet an outside militarized government, whose officials are interested in understanding and helping him.  This other city's infrastructure appears to be state-of-the-art and well funded.  The officials all have nice uniforms.  However, their environment is also under the same influence that's making everyone see things differently than what they really are.  The main character somehow sees through this illusion too, and realizes this fake prosperity is affecting others outside his home town--maybe the entire world!    The movie ends with       the man and woman eventually running into some elderly maintenance man in a forested area.  The maintenance man's job is to service a machine that alters everyone's perception of reality.     I posted about this a couple of years ago on another site, and somebody took a guess that it was "The Prisoner" but unfortunately that's not it.  I didn't really explain it well the first time, so hopefully this is clearer. If diary Tom Riddle had succeeded, would there have been two Voldemorts? I came across the following question:     If 3 boys can paint a fence in 2 days what part of the job can     be completed by two boys in 1 day?   The answer to this problem is $\frac13$.  I can't manage to tackle this problem. Could anyone tell me what would be the quickest way to solve this problem (This question is a standardized test based question). Any help would be appreciated. Given two equivalent norms, convergence in one implies convergences in other. Suppose we have two equivalent norms, $\|\cdot \| \sim \left\vert\!\left\vert\!\left\vert \cdot\right\vert\!\right\vert\!\right\vert$. We assume that $x_n \rightarrow x $ in $(X,\|\cdot \|)$. Show that $x_n \rightarrow x $ in $(X,\left\vert\!\left\vert\!\left\vert \cdot\right\vert\!\right\vert\!\right\vert)$  We observe since the norms are equivalent, there exits constants $\alpha, \beta$, such that $\alpha \|\cdot \| \leq \left\vert\!\left\vert\!\left\vert \cdot\right\vert\!\right\vert\!\right\vert \leq \beta \|\cdot \|$, with $0 \lt \alpha \leq \beta$  Since, $x_n \rightarrow x$ in $(X,\|\cdot \|$) we know that for all $\epsilon&gt;0, \exists K(\epsilon)$ such that for n>$K(\epsilon),$ we have $\|x_n -x\| \leq \epsilon$  Since, we know $\beta \gt 0$ , then $ \beta \|x_n -x\| \leq \beta \epsilon$, Thus we see that $\left\vert\!\left\vert\!\left\vert x_n-x\right\vert\!\right\vert\!\right\vert\leq \beta \epsilon$.  This is how far I've gotten, our goal to is to find a a $J(\epsilon)$ such that for all $\epsilon&gt;0, \exists J(\epsilon)$ such that for n>$J(\epsilon),$ we have $\left\vert\!\left\vert\!\left\vert x_n-x \right\vert\!\right\vert\!\right\vert \leq \epsilon$  My candidate for $J(\epsilon$) is $J(\epsilon) = \frac{K(\epsilon)}{\beta}$ from my above argument, but I'm not exactly sure how to validate this. If someone could help me on the proof, it would be very appreciated. I believe I'm pretty close, just missing something small. How to clone ArrayList and also clone its contents? I want it to know how to convert string field values to integer values in QGIS Dufour... there are numbers and strings in the same field, when I use toreal(string) it stops and it won´t create the field. Compact operators on an infinite dimensional Banach space cannot be surjective I am reading a book about functional analysis and have a question:      Let $X$ be a infinite-dimensional Banach-space and $A:X \rightarrow X$ a compact operator. How can one show that $A$ can not be surjective? In classical statistics, there is a definition that a statistic $T$ of a set of data $y_1, \ldots, y_n$ is defined to be complete for a parameter $\theta$ it is impossible to form an unbiased estimator of $0$ from it nontrivially. That is, the only way to have $E h(T (y )) = 0$ for all $\theta$ is to have $h$ be $0$ almost surely.   Is there a intuition behind this? Is seems like a rather mechanical way of defining this, I am aware this has been asked before, but was wondering if there was a very easy to understand intuition that would make introductory students have an easier time digesting the material.