Let $(a_n)$ be an unbounded strictly increasing sequence of positive real numbers and let $x_k=\frac{a_{k+1}-a_{k}}{a_{k+1}}$. Then I want to find the correct option (s).   For all $n\geq m, \sum\limits^{n}_{k=m}x_k&gt;1-\frac{m}{n}$ For all $n\geq m, \sum\limits^{n}_{k=m}x_k&gt;\frac{1}{2}$ $\sum\limits^{\infty}_{k=1}x_k$ converges to a finite limit $\sum\limits^{\infty}_{k=1}x_k$ diverges to $\infty$   My observation says that option 2 is false and so is 3. But I couldn't conclude anything for 1 and 4. Please help!
So I've been thinking through some test cases. If $a_n = n$ then $\sum_n \frac{a_{n+1} - a_n}{a_n}$ is the harmonic series which diverges. And if $a_n = \sum_{k=1}^n 1/k$ then $\sum_n \frac{a_{n+1} - a_n}{a_n}$ diverges like $\sum_n 1/(n \log n)$. So that got me thinking, if $a_n$ is a strictly increasing unbounded sequence, does $\sum_n \frac{a_{n+1} - a_n}{a_n}$ necessarily diverge?
In Russian we have idiom/saying "To shoot out of cannon into sparrows" (literal translation) which is used to convey an idea of applying too drastic measures to small problems. I believe there should be some native-English equivalents to this saying. Can you share if there are any? 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. Should one create slides similar to those that one uses in a good powerpoint presentation? Or are there things that a poster should include that a powerpoint should not include? (and vice versa) What is this type of flexible contact strip called? The LCD below is from a cheap digital clock which was (obviously) damaged (broken glass).    The electrical connection between the PCB and the display was realized with this pink strip.    Its flexible...    ...and consists of alternating rows of conductive and insulating material between the pink boarders.    [above image was altered in contrast/brightness to highlight the alternating strips]  What is this kind of connector called? Is it only used for displays?   I am pretty impressed that the connection is reliable just from the very slight pressure between the strip and the PCB. Also, its easily removable from the display. There seems to be some conductive layer on the glass, from where the signal goes into the displays internals. This connection seems to be damaged for the three rightmost signals, where the glass is broken. Can a Linux user belong to more than one group?  If it is allowed, then one use may access files from two groups, that would be quite nice!  If not, is there any alternative way to have this function?  This issue is not the same as  or How do I use SQL's GETDATE() and DATEADD() in a Linq to SQL expression? If I have a Linq to SQL expression like this:    from subscription in dbContext.Subscriptions  where subscription.Expires &gt; DateTime.Now select subscription   I want this to to use the SQL Servers GETDATE() function instead of the time of the machine running the C# program.  The next question would be how to translate this:  DateTime.Now.AddDays(2)   to this:  DATEADD(dd, 2, GETDATE()) This is a homework question of mine that Ive answered and I'd really like some feedback on my solution.       Prove or find a counter example to the claim that for all sets $A,B,C$, $$\text{if  }A\cap B=B\cap C=A\cap C=\emptyset \text{ then } A\cap B\cap C\neq \emptyset$$.   Since I can only either prove it is true, OR find a counter example I have gone with the latter because I think this statement is not true. My counter example is as follows:  Let $A = \{1,2,3 \}$ , $B = \{4,5,6 \}$ and $C = \{7,8,9 \}$. It is then clear that $A ∩ B =  B ∩ C =  A ∩ C = \emptyset$ .  If we let $x ∈ A\cap B\cap C$ then $x$ must be in $A$ and $B$ and $C$. Since this cannot be possible because there are no common elements then this implies that $A∩B∩C$ has to be equal to a null set i.e $A∩B∩C = \emptyset$ and thereby showing that (at least for this scenario) if $A \cap  B =  B \cap  C =  A \cap  C = \emptyset$ then $A\cap B\cap C = \emptyset$.  Is this okay? Any feedback will be greatly appreciated thank you! I committed and joined  in private beta. As per private beta privilege page casting close and reopen votes requires only 1 reputation. So anyone is able to access close votes and reopen votes review queues. But there is no link of the review page in the top bar or anywhere. So I thought I am not able to access any review queue. I opened the review page by typing the URL manually (to see review statistics for curiosity). But I found that I can access close &amp; reopen votes review queues.        Probably I won't be able to access any review queue if I have less than 350 and the site goes to public beta. But during private beta off topics, unclear, or closable questions are likely to come. So it is very important to close/reopen them during private beta.  So can we please add the "review" link in the top bar if I am able to review some queues? Can the 5E abjurer wizard recover temp hp on his arcane ward any time it is below max or only when it reaches 0? On p115 of the PHB under Arcane Ward there are a few facts that I think are clear:   The ward is created by the first abjuration spell you cast in a day It has a max hp of level &times; 2 + int modifier. The ward does not absorb damage once it reaches 0 hp, but it remains until you take a long rest The ward can be recharged by casting abjuration spells, at a rate of 2 x spell level worth of hp 3.   However, the way they wrote the paragraph about recharging seems unclear to me. Can you recharge the ward whenever it is below max hp or only when it is at 0? My understanding of temporary hp would say it is the latter, but the writing on this ability infers the former.  An example.  If I were a level 10 abjurer with 20 int and cast Mage Armor, it would give the ward 25 hp. If I took 30 damage from an enemy, the ward would absorb 25 damage, leaving it at 0, and I would take the remaining 5 damage. If I cast Stoneskin (level 4 abjuration) the ward would be recharged with 8 hp. If I then cast Planar Binding (level 5 abjuration) which would be correct?   The ward remain at 8 hp The ward would recharge an additional 10 hp to a total of 18 hp So, floor is a function that converts a real number to an integer. It rounds down. This makes sense; however, what about complex numbers? I know that depending on the number, it can be split linearly. However, I do not know what $\lfloor i\rfloor$ equals. Is there a clearly defined definition of $\lfloor i\rfloor$? How do I install applications in Ubuntu? What are these notices above my post? My post has been marked with an informational notice in a box above it.   What should I do about this? Who attaches notices to posts? What kinds of notices are available? What should I do if I think a notice should be added to a post? What should I do if I think a notice should be removed from a post? I am trying to find the name of a direct to TV movie, possibly a 2-3 part miniseries which I believe aired on the SciFi channel in the 90s. I don't remember who starred in it, but the movie opened with an astronaut attempting reentry whose shuttle ends up exploding. He awakes in a large bubble of air in a river and after reaching the surface and swimming to shore, meets several other people, from seemingly many different times throughout history (including one alien, who died on Earth).  Over the course of the movie, we learn the world is the afterlife, where everyone goes when they die. The major plot involves launching a steamboat (built by Mark Twain, no less) to travel up a river to the mountains, where the beings responsible for bringing everyone to this world are.  Does anyone know what the title of this movie/miniseries is? Illustrator: Is there an efficient and accurate way to join two circles by their tangents (and keep it editable)? It's easy to join two circles if they are the same size and horizontal/vertical, into a pill shape:    If the circles are at an add angle then it's still possible but you can no longer rely on smart guides.     You can add lines roughly where the tangents should be and then use the scissors tool on the circles, cut out the inner area and then join the two crescents into one shape.    Is there a better way to do this? Is there a way to do this that is still editable afterwards too?  There are some other questions that cover the same area as this but don't cover editability or joining circles that are not in line with each other (where smart guides can't help). I need to clip a vector layer to the extent of a raster. It has to be very precise that's why I would not want to draw a polygon myself (and clip the two polygons). There seem to be some ways to clip a raster to a polygon but I have difficulties to find a solution to clip a polygon to a raster. I already tried to create a polygon out of my raster several times in ArcGIS and R, so I could clip the result. But every time it crashes because the raster is to big. There must be a simple way to solve this, and somehow I can just not see it at the moment.  How can I do this using GDAL? Fortunately, I have very few strongly negative interactions with cars (most people generally seem to respect the way I ride), but those that do stick with me for a while.  The scenario is almost always the same. Rude driver approaches, does their rude thing (obscenities, "get off the road," unsafe passing, etc...), then speeds off to annoy the next bicyclist (or more than likely arrives at the next red light).  While this is generally a short interaction, I find that I just don't have a good rest of the ride. I find myself angry at the driver, thinking about what I could have done differently, and just generally being pissed off. Clearly I realize that it is possible I'm a jerk, but given the relative infrequency of these events I'm hoping that's not the case and I'm just riding defensively and this driver is having a bad day.  What can I do to get over (or mitigate) my anger at inconsiderate drivers? Prove that $\lim_{x \to \infty}\big(\frac{x}{x-1}\big)^x$ is also $e$. I have two atoms, both of which are either bosons or fermions, with four allowed energy states: $E_1 = 0$, $E_2 = E$, $E_3 = 2E$, with degeneracies 1, 1, 2 respectively.  What's the difference between the partition functions of a pair of two bosons and that of a pair of two fermions? Migrated questions with a nonexistent author should still show the author's name in the questions list I just noticed . Its author doesn't have an account on this site, which makes it show up like this on the homepage:        It looks like a simple thing - but it gets in the way of the 'flow' of the page:        Couldn't we somehow get the author's name, so that migrated questions with authors that don't have an account here still show up, like this:        Much better. Please? Say I have the following CSS and HTML code:    #header {   height: 150px; } &lt;div id="header"&gt;   &lt;h1&gt;Header title&lt;/h1&gt;   Header content (one or multiple lines) &lt;/div&gt;     The header section is fixed height, but the header content may change.  I would like the content of the header to be vertically aligned to the bottom of the header section, so the last line of text "sticks" to the bottom of the header section.  So if there is only one line of text, it would be like:   ----------------------------- | Header title | | | | header content (resulting in one line) -----------------------------   And if there were three lines:   ----------------------------- | Header title | | header content (which is so | much stuff that it perfectly | spans over three lines) -----------------------------   How can this be done in CSS?