Do there exist sequences $\{a_n\}$ and $\{b_n\}$ satisfying all of the following properties?   $a_n&gt;0$ and $b_n&gt;0$ $\{a_n\}$ and $\{b_n\}$ are both decreasing $\sum a_n$ and $\sum b_n$ both diverge $\sum\min\{a_n,b_n\}$ converges
Find examples of two series $\sum a_n$ and $\sum b_n$ both of which diverge but for which $\sum \min(a_n, b_n)$ converges. To make it more challenging, produce examples where $a_n$ and $b_n$ are positive and decreasing.  Edit: This problem is taken verbatim from Exercise 2.7.11 on page 68 of Abbott's Understanding analysis.
Line-breaking widget layout for Android I'm trying to create an activity that presents some data to the user. The data is such that it can be divided into 'words', each being a widget, and sequence of 'words' would form the data ('sentence'?), the ViewGroup widget containing the words. As space required for all 'words' in a 'sentence' would exceed the available horizontal space on the display, I would like to wrap these 'sentences' as you would a normal piece of text.  The following code:  public class WrapTest extends Activity {     /** Called when the activity is first created. */     @Override     public void onCreate(Bundle savedInstanceState) {         super.onCreate(savedInstanceState);         LinearLayout l = new LinearLayout(this);         LinearLayout.LayoutParams lp = new LinearLayout.LayoutParams(                 LinearLayout.LayoutParams.FILL_PARENT,                 LinearLayout.LayoutParams.WRAP_CONTENT);         LinearLayout.LayoutParams mlp = new LinearLayout.LayoutParams(                 new ViewGroup.MarginLayoutParams(                         LinearLayout.LayoutParams.WRAP_CONTENT,                         LinearLayout.LayoutParams.WRAP_CONTENT));         mlp.setMargins(0, 0, 2, 0);          for (int i = 0; i &lt; 10; i++) {             TextView t = new TextView(this);             t.setText("Hello");             t.setBackgroundColor(Color.RED);             t.setSingleLine(true);             l.addView(t, mlp);         }          setContentView(l, lp);     } }   yields something like the left picture, but I would want a layout presenting the same widgets like in the right one.        Is there such a layout or combination of layouts and parameters, or do I have to implement my own ViewGroup for this? Prove that $\displaystyle \sum^{n}_{r=0}(-1)^r\cdot \large\frac{\binom{n}{r}}{\binom{r+3}{r}} = \frac{3!}{2(n+3)}$   $\bf{My\; Try::}$ We can write $$\frac{\binom{n}{r}}{\binom{r+3}{r}} = \frac{n!}{r!\times (n-r)!}\times \frac{r!\times 3!}{(r+3)!} = \frac{3!\times n!}{(n-r)!\times (r+3)!}$$  So our sum is $$ 3!\sum^{n}_{r=0}(-1)^r\frac{n!}{(n-r)!\times (r+3)!} = \frac{3!}{(n+1)(n+2)(n+3)}\sum^{n}_{r=0}(-1)^r\frac{(n+3)!}{(r+3)!\times (n-r)!}$$  So we get Sum $$ \frac{3!}{(n+1)(n+2)(n+3)}\sum^{n}_{r=0}(-1)^r\binom{n+3}{r+3} = -\frac{3!}{(n+1)(n+2)}\sum^{n}_{r=0}(-1)^{r+3}\binom{n+3}{r+3}$$  $$  = -\frac{3!}{(n+1)(n+2)(n+3)}\left[-\binom{n+3}{3}+\binom{n+3}{4}-\binom{n+3}{5}--.......+(-1)^{n+3}\binom{n+3}{n+3}\right]$$  $$=-\frac{3!}{(n+1)(n+2)(n+3)}\left[\binom{n+3}{0}-\binom{n+3}{1}+\binom{n+3}{2}-\binom{n+3}{3}+\binom{n+3}{4}-\binom{n+3}{5}--.......+(-1)^{n+3}\binom{n+3}{n+3}\right]+\frac{3!}{(n+1)(n+2)(n+3)}\left[\binom{n+3}{0}-\binom{n+3}{1}+\binom{n+3}{2}\right]$$  So we get $$ = \frac{3!}{(n+1)(n+2)(n+3)}\left[1-(n+3)+\frac{(n+3)(n+2)}{2}\right] = \frac{3!}{(n+1)(n+2)}\times \frac{n^2+3n+2}{2} = \frac{3!}{(n+1)(n+2)(n+3)}\times \frac{(n+1)(n+2)}{2} = \frac{3!}{2(n+3)}$$  Can we prove it Using Combinatorial way,bcz above method is very lengthy  If yes then plz explain here  Thanks Rules and guidelines for drawing good schematics There are a lot of poorly drawn schematics here. A few times people have actually asked for critiques of their schematics. This question is intended as a single repository on schematic drawing rules and guidelines can point people to. The question is  What are the rules and guidelines for drawing good schematics?     Note: This is about schematics themselves, not about the circuits they represent. How do I restore the sources.list file? How can I use JUnit4 idiomatically to test that some code throws an exception?  While I can certainly do something like this:  @Test public void testFooThrowsIndexOutOfBoundsException() {   boolean thrown = false;    try {     foo.doStuff();   } catch (IndexOutOfBoundsException e) {     thrown = true;   }    assertTrue(thrown); }   I recall that there is an annotation or an Assert.xyz or something that is far less kludgy and far more in-the-spirit of JUnit for these sorts of situations. For users running Java application or applets in the browser, what difference does it make running either Sun Java or OpenJDK?  EDIT  Oracle will be , and basing the proprietary implementation on OpenJDK. Can we expect similar experience in running Java apps using either OpenJDK or Oracle JDK? What does the 'static' keyword do in a class? How to revert uncommitted changes including files and folders? Is there a way to call process builder from Apex? Prove $e^x - e^y \leq e |x-y|$ for $x$ belonging to $[0,1]$ I'm not sure how to go about this. Does it involve using MVT?  I got as far as saying $e = \frac{e^x - e^y}{x-y}$. Why does cold metal seem colder than cold air? (I apologize for this elementary question. I don't know much about physics.)  Let's say that I put a metal pot in the refrigerator for several hours.  At this point, I guess, the pot and the air (in the refrigerator) have the same temperature.  Now, I touch this pot. It feels very cold. But when I "touch" the air (that is inside the fridge) it doesn't "feel" as cold. I don't feel the same "ouch!" that I feel when I touch the pot.  Why is that? Why does the metal seem colder than air although they both have the same temperature?  (I know that gas has less particles in it in one unit of volume compared to solids and liquids, but since "temperature" means "the average kinetic energy", these fewer air particles are supposed to hit my hand in a velocity that's going to compensate for their lower number, aren't they?)  A related question, for clarification:  If I use a thermometer to measure the temperature of the pot &amp; air (let's assume it's a thermometer that has a probe that can touch objects), will it show the same reading for both? If so, what makes the thermometer different than my hand? I mean, my hand is sort of a thermometer, so why would it fail whereas a non-human thermometer would work? Can someone please explain how I can install  in the following ways?   By using the package manager? By downloading and installing a pre-built package? Proving a function is constant, under certain conditions? The problem:  Assume $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfies $|f(t) - f(x)| \leq |t - x|^2$ for all $t, x$. Prove $f$ is constant.  I believe I have some intuition about why this is the case; i.e. if $t$ and $x$ are very close, ($|t - x| = \epsilon$), then $\sqrt{\epsilon}$ will converge to 0, so $|f(t) - f(x)|$ will also converge to zero if you keep taking $t$ and $x$ closer and closer to each other. However, how do I formalize this argument? Thanks. "some " vs. "some of " Some of rules can be the same for most websites.   vs.     Some rules can be the same for most websites.   I think I want to say     Some of these rules can be the same for most websites. How can author deal with late reviews? Simple bijection $\mathbb N_0 \times \mathbb N_0 \to \mathbb N_0$ I'm looking for a 'simple' bijection $$ \pi \colon \mathbb N_0 \times \mathbb N_0 \to \mathbb N_0, $$  where 'simple' in this context means that it should be as easy as possible to define and it should be self-evident that said function is in fact a bijection. It's a quest for convenience - not minimal complexity in any rigorous sense.  Here are a couple of examples I came up with - none of which satisfy both requirements:   Let $\pi(m,n) = c(2^m3^{n+1})$, where $c \colon \{2^m3^{n+1} \mid m,n \in \mathbb N_0 \}\to \mathbb N_0$ is the order isomorphism (or in fact Mostowski collapse) of its domain under the natural ordering, Let $\pi(m,n) = \langle n,m \rangle$ - Gödel's pairing function, Let $\pi(m,n) = m \oplus n$ - the number resulting from '' $m,n$ (where we imagine both $m$ and $n$ as in infinite decks of cards whose $i$th element is labeled with its respective $i$th digit). ... What if I offer a bounty and don't get any good answers? I have started a bounty:   It expires in 3 days. I don't have an appropriate answer to accept, and the only answer which it has gotten has no votes.  So am I going to simply lose 100 reputation without getting a satisfactory answer?  Or should I accept the lone inappropriate answer so that I will not lose my right to accept answers to any questions which I might ask in the future, which is what the bounty rules claim? Atmospheric pressure experiment using a cup with a fluid to hold a glass plate When I was in high school, my teacher did an experiment to show the power of atmospheric pressure.  Experiment: Prepare a glass bottle, fill with water, put a glass plate on the bottle, make sure there is no air in the bottle, just water. Hold the glass bottle and plate, and invert them. Slowly, release the hand which hold the plate. Because the atmospheric pressure, the glass plate will not fall down.    Question 1: As we know, if we didn't put water into the glass bottle, just air in the glass bottle the glass plate will fall down. So, If we use other liquid instead of water, this liquid has smaller density than water, like alcohol, the plate will not fall down. But, if we continue to chose the smaller density liquid until the liquid has density like air the plate will fall down. So, if there is a threshold density of liquid make the plate between falling  down and not falling down?  Question 2: If we put the bottle and plate into water slowly, when the plate and part of bottle are in the water but part of the bottle still above the water, will the plate fall down?  Question 3: Continuing question 2, when both bottle and plate are in the water I know the plate will fall down. So, how does the pressure of water and atmosphere change?  Is there a good description of this phenomena?  Why does the trick continue to work with lower density fluids and why does placing the plate in the water cause it to fall? I'm trying to get Intel and Nvidia graphics card working together with two monitors. One monitor should use the Intel card and the other the Nvidia card. Of course, I want to use only one keyboard and one mouse to switch between both screens. Bumblebee didn't work for me.  I don't want to use the xserver-xorg-video-all package because I need the Nvidia card for gaming and I can't use only one graphics card because I need one for KVM virtualisation and the other one for my Linux desktop  So I tried to install the drivers and to configure the Xserver manually. The Nvidia driver is working and I get a video output. But I don't get a video output with the Intel card.  I followed   The config files: There are a lot of legendaries available in ORAS, however a lot of them will only show up/be available if certain prerequisites are met.  I know Groudon/Kyogre are available through the Story, and      Rayquaza &amp; Deoxys   through the postgame Delta episode, but what about the other available legendaries? What do I need to do to catch them?