Show that if series $a_n$ converges then series $(-1)^n a_n$ converges.    Is that true? A quick proof is highly appreciated.Thanks
I have this statement and I need to say if it is true or false:  Let $\{a_n\}$ be a real sequence.  $$\lim_{n\to +\infty} a_n = 0 \quad \implies \quad \sum_{n=1}^{\infty}(-1)^na_n \quad \text{converges}$$  I know, from the Leibniz criterion that:     If $a_n \to 0$, $a_n$ is decreasing and positive then $\sum_{n=1}^{\infty}(-1)^na_n$ converges   From this fact I believe that the statement is false but I couldn't come up with an infinitesimal sequence that isn't decreasing and for that reason is divergent. I tried some function with $sin(\frac{1}{n})$ without any luck. Any help would be very appreciated, thank you!
Do I need Intel Smart Response when installing Ubuntu? After seeking clarification about linear model coefficients  I have a follow up question concerning non-signficant (high p value) for coefficients of factor levels.  Example: If my linear model includes a factor with 10 levels, and only 3 of those levels have significant p values associated with them, when using the model to predict Y can I choose to not include the coefficient term if the subject falls in one of the non-signficant level?  More drastically, would it be wrong to lump the 7 non-significant levels into one level and re-analyze? I've noticed that with the new question asking form, sometimes, attempting to submit a question will post two identical copies of the same question.  I've  (), but at the time, I assumed it was just a temporary fluke. As far as I remember, I just reviewed the question, saw no errors, and submitted it. (That's all the info I have.)  However, I just saw this happen to someone else, so it can't just be a fluke on my end. , and . The author claims there that they did indeed double click the "Review" button, but their question didn't go through since it didn't have a required tag, but when they tried to submit the question with the required tag, they only clicked it once.  Also, now that I think about it, it could not have been just a temporary client-side issue on both our ends. The system is supposed to  if one tries to post more than one question in 30 seconds. Neither of us got a CAPTCHA. This means that the server accepted the double post, and so it's a server issue.  This has the effect of denying users badges that they would have otherwise qualified for, since a single negatively-received, closed, or deleted question . This means that even if the double post is deleted and the original is upvoted, the user won't qualify for that day since they've made a "bad" question on that day. (While they may be able to later reverse this 60 days later by undeleting and re-deleting their question - questions deleted more than 60 days later aren't considered "deleted" for the check - they often won't be able to, since most of the time, users will have downvoted the double post, which also disqualifies one from the badges.)  Can an employee go through server logs to see what happened in both these cases, and add some safeguards to prevent accidental double submissions of questions? This seems to have come up when the new asking form was deployed.  Update: There was another case just now:  was . As both have the exact same time of submission, this is an instance of this bug, since as I said above, this thing is ordinarily prevented by a CAPTCHA. $P[X=x]=0$ when $X$ is continuous variable What determines which frames are inertial frames? I understand that you can (in principle) measure whether "free particles" (no forces) experience accelerations in order to tell whether a frame is inertial. But fundamentally, what determines which frames are inertial (i.e. what principle selects in which frames free particles will not appear to accelerate)? I've been told that the cosmic microwaves determine the ultimate rest frame of the universe, but that doesn't make sense to me, since one can still ask why that frame is an inertial frame.  Also, I understand that there are no real inertial frames in general relativity, but it seems like there certainly exists approximate inertial frames and we can ask why those frames are approximately inertial and not others. For example, in the frame of a person riding a merry go round, free particles appear to greatly accelerate; while in the frame of someone standing next to the merry go round there are no such great accelerations. Why does the guy (or gal) on the merry go round see free particles accelerating while the other guy doesn't.   And if you're gonna tell me that it's "the rest of the stuff in the universe" that determines whether the person on the merry go round sees free particles accelerate, I'll ask how you know that all that stuff is not spinning.  I hope this question sort of makes sense, it's been bothering me for a while and my study of relativity (most of special relativity and just the outline of general relativity) hasn't really clarified it for me much. Value of Vandermonde type determinant I'm trying to create record Type selection screen in LWC, but getting this error  afterRender threw an error in 'c:createContact' [Cannot read property 'recordTypeInfos' of undefined]  HTML:  &lt;template&gt;     &lt;lightning-combobox name="recordType" label="Record Type" placeholder="Choose Status"                         value={value}                          options={recordTypeId}&gt;      &lt;/lightning-combobox&gt; &lt;/template&gt;   JS:  import { LightningElement, track, wire, api } from 'lwc'; import { getObjectInfo } from 'lightning/uiObjectInfoApi'; import CONTACT_OBJECT from '@salesforce/schema/Contact';  export default class CreateContact extends LightningElement {      @track statusOptions;     @track value;     @api objectApiName;      @track objectInfo;      @wire(getObjectInfo, { objectApiName: CONTACT_OBJECT,})     objectInfo;       get recordTypeId() {         // Returns a map of record type Ids          console.log(this.objectInfo);         const rtis = this.objectInfo.data.recordTypeInfos;         return rtis;     }  } To change URL to user friendly URL I'm refactoring my ASP.net application from ASP.net 3.5 to 4.0. Also I'm changing the URLs to user friendly URLs /product.aspx?id=100 to /product-name/100.  All my pages are indexed by search engines and the site has already been online for 6 years. I'm planning to do 301 redirect from the old pages to the new one.  I want to make sure I won't lose the rankings and traffic.  Any suggestion how to do it properly? If $p$ is congruent to $2 \pmod 3$, how can I prove that all $a$, $1 \le a \le p-1$ are cubic residues $\mod p$?  Here's what I've done:  $1^3$ congruent to $1 \pmod p$ thus, 1 is a cubic residue, Also $(p-1)^3=(p-1) \pmod p$ implies  $$(p-1)^3 - (p-1) = (p-1)[(p-1)^2 -1] = (p-1)(p^2 -2p) = p(p-1)(p-2) $$ implies $$p|(p-1)^3 -(p-1)$$ Thus  $p-1$ is a cubic residue $\mod p$. Now I don't know how to show for $1\lt a \lt p-1$ are all cubic residues when $p$ is congruent to $2\pmod 3$. Should an image carousel on a mobile site automatically rotate? Or should the user have complete control? The intention is to have a slider/carousel that shows 6 or 7 images which the user can look through by clicking arrow buttons left or right. But the question has also come up if the user has not started looking through the images, should the carousel start automatically rotating the images. Since this is for a mobile site, I'm uncertain... Is binary logistic regression a special case of multinomial logistic regression when the outcome has 2 levels? Show that   $$\displaystyle1 - \frac{x^2}2\leq\cos x\quad\forall x \in \mathbb{R}$$   Let $f(x) = \cos x - 1 + \frac{x^2}2$; then we need to show that $f(x) \geq 0\quad\forall x \in \mathbb{R}$.  Since $$\lim_{x \to \pm\infty} f(x) = +\infty$$ and $$\operatorname{im}f'' = [0, 2]$$ (as $f''(x) = 1 -\cos x$), it remains to show that $x = 0$ is a minimum for $f(x)$, but I cannot do that because I would need to solve $f'(x) = x -\sin x = 0$, which I cannot. I think it would suffice to show that plugging in $x = 0$ we have that $f'(x) = 0$, and combining it with the fact that the function $f$ is convex, we would be done.    Is it enough? What's a better proof?  EDIT: It just occurred to me that there is a way to solve $f'(x) = 0$. Just observe that $f'(x)$ is an odd function and that $f''(x) \geq 0$ implies that $f'(x)$ is monotonically increasing. Since an odd function, if defined at $x = 0$, must be $0$ there, $x = 0$ is a solution to the equation and it's the only one. The set of real numbers and power set of the natural numbers I have learnt that the cardinality of the power set of the natural numbers is equal to the cardinality of the real numbers. What is the function that gives the one-to-one correspondence between these two sets?  I have also learnt that there exists no set whose cardinality is strictly between the natural numbers and the real numbers. Is there a proof of this or at least some intuitiveness behind it? How can I have a "Dear %firstname%" template in my Gmail? It seems to be a very easy thing but I am struggling to see how. I need a "Dear %firstname%,  [cursor here] Kind Regards etc" template in my Gmail account, so every time I click reply, it is already there. How can I do that? Is there a well-known term for the synonym or near-synonym "telescoping" words? This has been rattling around in the back of my mind for many years (way before Stack Exchange came into existence), so it's a relief to finally ask the question.  There are words that can be "contracted" by removing one or more letter(s) (without changing the order of the letters) to form a shorter word with the same meaning, or one very close to it.  The two strongest examples I can think of right now are:  rapscallion -> rascal  and  satiate -> sate  This is a slightly weaker example:  rapine -> rape  Even though "rape" has come, in modern usage, to refer almost exclusively to a sexual crime, it can still be used in a more literary (and purposefully anachronistic) sense to mean "the plunder and pillage of a country or region". In that sense, "rape" can be used in a similar fashion to "rapine"; in any case, the two words are etymologically related to the Latin rapere which is tied to the archaic usage.  Another weaker example:  transliterate -> translate  These words are near-synonyms, but there is a definite difference in their meaning. Still, I guess this pair would qualify, albeit in a looser sense.  I wanted to exclude some relatively trivial examples, such as:  (inflammable -> flammable)  (perambulate -> ambulate)  and so forth. So I've imposed the additional rule (arbitrary though it may seem) that the "stripped" letters must not come from the start or the end of the word, but only from its middle.  Acronyms and portmanteaus are also excluded, as are "slangier" contractions e.g.:  (modulator-demodulator -> modem)  (motor-hotel -> motel)  (certitude or certainty or certificate -> cert)  I guess the long-short forms of the same word would form a linguistic doublet. I've coined the term "telescoping word" to describe the long form that "collapses" into the short form with letter deletion sans anagramming.  My question is: has this phenomenon already been studied? If so, what name have authorities given it? Is there already an existing reference or resource on it? Thanks in advance.  (This is not a duplicate of , as that is asking about the more general phenomenon of words (not necessarily synonyms) occurring within larger words. In fact, the question specifically mentions "kangaroo word" and states that a broader scope is being sought.) Using C# .NET 2.0, I have a composite data class that does have the [Serializable] attribute on it.  I am creating an XMLSerializer class and passing that into the constructor:  XmlSerializer serializer = new XmlSerializer(typeof(DataClass));   I am getting an exception saying:      There was an error reflecting type.   Inside the data class there is another composite object.  Does this also need to have the [Serializable] attribute, or by having it on the top object, does it recursively apply it to all objects inside? past simple or present perfect Please have a look at this sentence:     We haven't gotten a lot of Finnish groups that we liked here at Bananas but I'm pretty happy with this four tracker from Helsinki, apparently recorded in the same place where the Ramones played in the 70's.     The first verb is in present perfect (up to now) and the second is in simple past tense. I don't understand the simple past for the second verb.  Why "like" is not in its present perfect form as the first verb is? While playing Phase Ten, is it legal to pick up the top card of the discard pile and then immediately discard that exact card? How can I handle the warning of file_get_contents() function in PHP? Bind a key for multiple units to use the same spell