Why does the term in this  equal to infinity.  I thought when a term goes to zero the sum of all the terms added over a series goes to a specific value.   What do I miss?
If a sequence converges in a metric space, it is Cauchy, and in $\mathbb{R}^k$ every Cauchy sequence converges.   Therefore, in $\mathbb{R}^k$ a sequence converges iff it is Cauchy.  Let $\{s_n\}$ be a sequence in $\mathbb{R}$ where each $s_n=\sum_{k=1}^na_k$. Therefore, by the above, every series converges iff  $$\left | \sum_{k=m}^n a_k\right| &lt;\epsilon$$ For a given $\epsilon &gt;0$ and an integer $N$ such that $N\le m\le n$. If $n=m$ then the statement reduces to:  A series converges if and only if $$|a_n| &lt; \epsilon $$ For a given $\epsilon &gt;0$ and an integer $N$ such that $N\le n$.  This clearly cannot be (e.g Harmonic series). When does the equivalence become an implication.
Let $\{a_n\}$ be an unbounded, strictly increasing sequence of positive real numbers and let $x_k=\frac{a_{k+1}-a_{k}}{a_{k+1}}$. Which of the following statements is/are correct? (CSIR NET December 2014)         For all $n\geq m, \sum\limits^{n}_{k=m}x_k&gt;1-\bf{\frac{a_m}{a_n}}$   There exist $n\geq m, \sum\limits^{n}_{k=m}x_k&gt;\frac{1}{2}$      My attempt:  Analysing question in light of $a_k=k\implies x_k=\frac{k+1-k}{k+1}=\frac1{k+1}$, I feel both are correct, I doubt if  I can generalise the result in light of this one example.   Any way to prove that this true for general case?    Note: Please do not confuse  with my question. The first two options are different, I've marked the difference in bold. I don't have any knowledge in hardware and would like to know which RAM should I buy for my laptop.  How do I discover what RAM I need &amp; how much my machine can take?  I have a Razer Blade 15 (RZ09-02705F76) 2018 but don't know if there is a specific type or format of RAM that suits it and don't know where to find this information. Why does ( exit 1 ) not exit the script? Does an absolutely integrable function tend to $0$ as its argument tends to infinity? Suppose that $f:[0,\infty)\rightarrow\mathbb{R}$ is continuous. Is it true that   $$\int_{0}^\infty|f(t)|dt&lt;\infty\Rightarrow \lim_{t\rightarrow\infty}f(t)=0?$$  If so can you provide a proof, otherwise a counter example. Thank you. In the first book, Harry sees his Gringotts vault for the first time.  (Quotes thanks to Slytherincess!)     Griphook unlocked the door. A lot of green smoke came billowing out, and as it cleared, Harry gasped. Inside were mounds of gold coins. Columns of silver. Heaps of little bronze Knuts.      "All yours," smiled Hagrid.      All Harry's -- it was incredible. The Dursleys couldn't have known about this or they'd have had it from him faster than blinking. How often had they complained how much Harry cost them to keep? And all the time there had been a small fortune belonging to him, buried deep under London.      Hagrid helped Harry pile some of it into a bag. "The gold ones are Galleons," he explained. "Seventeen silver Sickles to a Galleon and twenty-nine Knuts to a Sickle, it's easy enough. Right, that should be enough fer a couple o' terms, we'll keep the rest safe for yeh."      Harry Potter and the Sorcerer's Stone - pages 58-59 - US Hardcover Edition   Jk Rowling posts in the Harry Potter Lexicon      JKR has stated in an interview (Comic Relief 2001) that she estimates the value of one Galleon to be "about five pounds," which works out to around US$9.75 (the exchange rate at the time of the interview was US$7.33). In the introduction to both QA and FB, US$250-million is stated to be the equivelent of 34 million Galleons. That also works out to a value of £5 to the Galleon, at the exchange rate of the time. The price listed on the back of the books, however, is not correct, since US$3.99 would equal less than half a British pound, or 8 sickles and 15 knuts. The book instead incorrectly lists US$3.99 as being equal to 14 sickles and 3 knuts. (Unfortunately, CNN.com uses this incorrect value for their Knuts-to-dollars converter.)      This gives us approximate values as follows:      1 bronze Knut = £0.01 (US about 2 cents)      1 silver Sickle = £0.29 (about US$0.57)      1 Galleon = £5.00 (more or less US$9.75)      These rates vary as the exchange rate fluctuates - see the Wizarding World Currency Converter for the current rate.   Now looking at the quote      Hagrid helped Harry pile some of it into a bag.   and     Right, that should be enough fer a couple o' terms, we'll keep the rest safe for yeh.   Obviously "some" is unspecified, but presuming the costs of ...     Typical elementary-school textbooks cost more than $100 each   He would be carrying at least 10 galleons for each school book, and going by the list of books just for his first year,  (Thanks Ibid!)     COURSE BOOKS      All students should have a copy of each of the following:         The Standard Book of Spells (Grade 1) by Miranda Goshawk   A History of Magic by Bathilda Bagshot   Magical Theory by Adalbert Waffling   A Beginners’ Guide to Transfiguration by Emeric Switch   One Thousand Magical Herbs and Fungi by Phyllida Spore   Magical Drafts and Potions by Arsenius Jigger   Fantastic Beasts and Where to Find Them by Newt Scamander   The Dark Forces: A Guide to Self-Protection by Quentin Trimble         (Harry Potter and Philospher's Stone, Chapter 5)   That's 80 Galleons just for books. This does not include the list of other expenses Harry has for his school years (wand, potion reagents, etc.)  How many Galleons is "some" or is Diagon Alley selling supplies for third world prices?  Sidenote, is this just a case of bad JK maths? SQL server Read Only permission automatically for new databases Need verification - Prove a Hermitian matrix $(\textbf{A}^\ast = \textbf{A})$ has only real eigenvalues Proof: Let eigenvalue $\lambda \neq 0$ such as  $$\textbf{A}\vec{v}  = \lambda\vec{v}$$ $$\Rightarrow (\textbf{A}\vec{v})^\ast = (\lambda\vec{v})^\ast$$ $$\Rightarrow (\vec{v}^\ast\textbf{A}^\ast)=(\lambda^\ast\vec{v}^\ast)$$ Right-multiply both sides by $\color{orangered}{\vec{v}}$$$\Rightarrow (\vec{v}^\ast\textbf{A}^\ast \color{orangered}{\vec{v}} )=(\lambda^\ast\vec{v}^\ast \color{orangered}{\vec{v}} )$$ $$\textbf{A}^\ast=\textbf{A}$$ $$\Rightarrow(\vec{v}^\ast\textbf{A}\vec{v})=(\lambda^\ast\vec{v}^\ast\vec{v})$$ $$\Rightarrow(\vec{v}^\ast\lambda\vec{v}) = (\lambda^\ast\vec{v}^\ast\vec{v})$$ $$\Rightarrow(\lambda\vec{v}^\ast\vec{v}) =  (\lambda^\ast\vec{v}^\ast\vec{v})$$ $$\Rightarrow \lambda = \lambda^\ast$$ $$\Rightarrow \lambda\in\mathbb{R}$$ Why do the measurements of this object seem erroneous? Black screen after suspend-after moving to the nvidia official drivers (not priority) Why does triply periodic univariate function not exist? A univariate function $f$ is periodic with period $p_1,\ldots,p_k$ if   $$f(z) = f(z + \sum_{i=1}^k n_i \cdot p_i)$$   for all complex $z$ and integers $n_i$. Elliptic function is an example of doubly periodic function.  that a triply periodic univariate function cannot exist, but why? Can't we follow the construction of elliptic functions that uses Schwarz-Christoffel formula to map upper-plane onto a triangle, and use the reflection principle to create a lattice which corresponds to three periods?     Which step fails to hold when we mimic the construction of an elliptic function to create a triply period meromorphic univariate function?   Sorry if the question is stupid. I do not have an access to the original proof that a triply periodic function cannot exist, either. So references are welcome! GetFeatureInfo from Image WMS Unfortunately MyApp has stopped. How can I solve this? I have this struct:  struct Map {     public int Size;      public Map ( int size )     {         this.Size = size;     }      public override string ToString ( )     {         return String.Format ( "Size: {0}", this.Size );     } }   When using array, it works:  Map [ ] arr = new Map [ 4 ] {     new Map(10),     new Map(20),     new Map(30),     new Map(40)};  arr [ 2 ].Size = 0;   But when using List, it doesn't compile:  List&lt;Map&gt; list = new List&lt;Map&gt; ( ) {     new Map(10),     new Map(20),     new Map(30),     new Map(40)};  list [ 2 ].Size = 0;   Why? PHP Codes are displaying in HTML I am running the latest version of X3 Terran Conflict (3.1) and after about 30 min - 1 hour it's memory usage goes well above 2 GB.  I only have 4 GB of ram, running Windows 7 64 bit.  As a result of X3 using more ram than is available, my hard drive starts paging and makes the game and other applications become unusable.  I have to restart the game.  Is anyone aware of this issue or how to fix it and is there any way in Windows or through some tool, to limit how much memory is given to the game?  Is this a bug in the game? Can this even be fixed?  I'm using some mods but I doubt any of them are causing this as they are simple mods like adding a tractor beam etc. Is it pythonic for a function to return multiple values? In python, you can have a function return multiple values.  Here's a contrived example:  def divide(x, y):     quotient = x/y     remainder = x % y     return quotient, remainder    (q, r) = divide(22, 7)   This seems very useful, but it looks like it can also be abused ("Well..function X already computes what we need as an intermediate value.  Let's have X return that value also").  When should you draw the line and define a different method? Formal basis for variable substitution in limits Are there integral solutions to the equation $x^2+y^2-z^2=1$? What does a de Broglie wave look like? What does a  wave look like? Are de Broglie waves transverse or longitudinal?   Can they be polarized?    What about the de Broglie wave of a ground state neutral spin-zero Helium 4 atom?   What experimental evidence do we have that supports the detailed nature of a de Broglie wave?     I have always assumed that de Broglie waves were mathematically  identical to electromagnetic waves, but I just realized there is no basis for this assumption, and in fact it must be false, unless there is an analogue to both the magnetic and the electric components of the electromagnetic wave.  So what does a de Broglie wave "look like"? I understood the very basic concept of how to use /etc/network/interfaces, but all I find online are examples, example after example, which I can copy-paste from. What I miss is an explanation of the syntax, an explanation of the meaning of the commands and which order the commands require. I want to understand, because most of the time copy-paste is not enough, because I'm not working on a fresh machine, so I can't just overwrite existing configurations because it would break a lot of stuff. man interfaces was not very helpful since it is written very complicated.  Example questions I have: what does inet in an iface line mean exactly (I could not even find it in the manpage), what does manual in an iface line mean exactly (many examples use it, but according to manpage it needs an extra config file then, which the examples don't present), when do I use or need them? When not? When I create a bridge, what exactly happens to the interfaces?