I do as follows  $(1+x)^n= \binom{n}{0}+\binom{n}{1}x+\binom{n}{2}x^2+...+\binom{n}{n}x^n$   differentiating in both sides  $n(1+x)^{n-1}= \binom{n}{1}+2\binom{n}{2}x+3\binom{n}{3}x^2...+n\binom{n}{n}x^{n-1}$  then, for $x=1$  $n \cdot2^{n-1} = \sum_{k=1}^{n}k\binom{n}{k}$  From here, I don't know how to proceed, I have tried to multiply both sides for $(1+x)$ to make $n \cdot2^{n}$ but then the summatory changes, I don't know how to prove the equality.
I am trying to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$ but am not able to think something except induction,which is of-course not necessary (I think) here, so I am inquisitive to prove this in a more general way.  The left side of an identity occurs while solving another problem (concerning binomial theorem) so I am more interested in deriving the right side from the left side, else I have to remember it now onward.  EDIT: I am more interested in an algebraic proof rather than combinatorial argument or something involving calculus (however I liked  and  solution), hence I am removing the combinatorics tag.
The Barbarian's Rage ability gives advantage on Strength checks and saving throws.  Does this mean that the Barbarian gets advantage to attack rolls as well? Is an attack a check, or since saving throws are mentioned, that means it does not apply to attacks? "Is far from..." - How do I connote the simile, without the literal connotation? Two identical files; Apache says one is forbidden Recovering eigenvectors from SVD Using matplotlib in pygame Are moving objects producing stronger gravity fields? If the strength of gravitational influence exerted by a body is derived from its mass and energy then is it true that a moving object which has some kinetic energy should also produce stronger gravitational pull than a static/slower one?  If that's correct then would it be also possible for an object moving fast enough to produce gravity so strong that it would collapse into a (fast moving) black hole? ART has better performance and better battery life on paper at least.  Should I consider switching to it on my moto g? What are the results I can expect? What precautions should I take? Is it easy to turn back to dalvik? Let's say you have the ability to shine some light into a perfectly round sphere and the sphere's interior surface was perfectly smooth and reflective and there was no way for the light to escape.  If you could observe the inside of the sphere, what would you observe? A glow? And would temperature affect the outcome?  Seems silly, it's just something I've always thought about but never spent enough time (until now) to actually find an answer. Find three Poisson-distributed random variables, pairwise independent but not mutually independent My friend wants to consolidate his various retirement account accumulated over the past 20-30 years to make it easier to manage/track his retirement savings. He doesn't recall immediately all the accounts he currently has (one of the reasons to rollover), but we've determined he has at least the following types of accounts: 401(a), 401(k), 403(b), 457(b), and "something from the city". He's trying to track down all the relevant paperwork for all of the accounts, but we'd like to know what kind of limits he might have to watch out for.   Are there any restrictions on the types of the accounts that can be rolled over? Is there a limit to the number of accounts that can be rolled into a single IRA (whether all at once or over the lifetime of the IRA)? Is there a limit to the number of times accounts can be rolled over (can/should he do 1 or 2 at a time as he gets the paperwork in order, or wait until he has everything sorted out)? Is this a yearly limit or lifetime? Is there a dollar limit on the amount that can be rolled over? Is there a limit to the age of accounts that can be rolled over?   As far as we know, all of the accounts are pre-tax/non-Roth accounts, and that Roth vs. traditional accounts should be rolled over to separate (Roth vs. traditional) IRAs. I'm using +[NSUserDefaults standardUserDefaults] to store application settings. This consists of roughly a dozen string values. Is it possible to delete these values permanently instead of just setting them to a default value? Are there any simple rules for choosing the definite vs. indefinite (vs. none) article? I can’t for the life of me figure out where to use a and where to use the — and where there is no article at all. Is there a simple rule of thumb to memorize?   The standard rule you always hear:     “If a person knows which item you are talking about then use "the"   . . . doesn’t clear things up for me, as I have no idea whether or not they know. Use of very as an adjective is (in my experience) most frequently attested in phrases like     ...the very person I was looking for.   To use adjective very with the indefinite article sounds quite bizarre in this case:     ?...a very person I was looking for.   But consider this sentence, where indefinite a is used:     Visualizing a very transformation of the master-slave relation, Thomas presses further: the slave is brother to the master, even doubly a brother.   OED treats the two uses as separate senses of the word, with the following definitions:           (first use):&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;With limitation (usually expressed by the or a possessive) to particular instances      (second use): Really or truly entitled to the name or designation   Here are two respective examples quoted in OED:     What would you say to me now, and I were your verie, verie Rosalind? (1616)   Thence we went into Queen Mary's room, and saw that beautiful portrait—that very queen and very woman. (1857)   Clearly, the difference in meaning between the two senses is quite light, and it seems to me like a kind of difference which is recurrent enough to have its own term. What is the proper term? Are there other English words having similarly related pairs of senses? And lastly, is OED jumping the gun in treating these as two senses? Could the difference in meaning instead be due to the choice of determiner? Usage in sentence:  Person A: Ooh - look at that crow!  Person B: Well, actually, as someone who is a scientist who studies crows, I am telling you, specifically, in science, no one calls jackdaws crows. They're not the same thing. If you're saying "crow family" you're referring to the taxonomic grouping of Corvidae, which includes things from nutcrackers to blue jays to ravens. It's not one or the other, that's not how taxonomy works.  Person A: Are you always such an XXX?   "Pedant" is close, but not quite.  It would be a mix of "know-it-all", "condescending" and "arrogant". Is it possible to adjust the frame rate of viewport playback? Is it possible to play an animation in the viewport at a slow framerate?  Just because of the rather heavy scene it's already going at about 8 fps, but I would like to be able to play it even slower (an in a controlled manner) for the purpose of recording an animation with automatic keyframes. Ubuntu 16.04 has been hanging on me ~1x per day.  This happens when I am in the middle of web browsing or using a desktop application, not when booting.  When it does, the mouse pointer will still move freely, but clicking or keystrokes have no effect on my system until I do a hard reboot.  What is the best way for me to debug this?  Here is some information:  selah@selah-Precision-Tower-5810:~$ uname -a Linux selah-Precision-Tower-5810 4.4.0-59-generic #80-Ubuntu SMP Fri Jan 6 17:47:47 UTC 2017 x86_64 x86_64 x86_64 GNU/Linux   Also, in case it is relevant, I have a "very big" monitor, a Dell 42" at 3840x2160 resolution.  selah@selah-Precision-Tower-5810:~$ lspci | grep VGA 03:00.0 VGA compatible controller: NVIDIA Corporation GM107GL [Quadro K2200] (rev a2)     UPDATE:  Following Artyom's advice I found the following message in my error logs:   Apr 27 09:47:25 selah-Precision-Tower-5810 kernel: nouveau 0000:03:00.0: fifo: SCHED_ERROR 0a [CTXSW_TIMEOUT] Apr 27 09:47:29 selah-Precision-Tower-5810 kernel: nouveau 0000:03:00.0: fifo: SCHED_ERROR 0a [CTXSW_TIMEOUT] Apr 27 09:47:33 selah-Precision-Tower-5810 kernel: nouveau 0000:03:00.0: fifo: SCHED_ERROR 0a [CTXSW_TIMEOUT]   Which has let me to this bug which describes similar behavior: When is it appropriate to use the original pronunciation of a foreign word versus the English pronunciation? When reading to an audience, or speaking in conversation, when is it appropriate to use the original pronunciation of a foreign word versus the English pronunciation (assuming you know the appropriate pronunciation for it)? Is it considered rude, or condescending? Or is it considered a mark of being knowledgeable?  One of the things to consider are place names. When referring to Paris, France, should it be pronounced with the silent 'S' as the French would say it, or with the 'S'? Should Hiroshima be pronounced as a Japanese speaker would pronounce it slightly more emphasis on RO, rather than the SHI? Should Mount Pinatubo be pronounced as a Filipino speaker would pronounce it with shorter stronger vowels, or the longer vowels? (e.g. Pi as is 'pick' rather than 'pea'.) Should Wichita be pronounces as the original "shi" rather than the modern "chi"?  Pronounce pesos or sombrero as a Spanish speaker 'eh' or the English 'ay'?  (I can't think of other common words right now that aren't words taken from other languages like hurricane, boondocks, tornado, etc. which I think have (correctly) changed to English pronunciations.) Differences in the behaviour of pinching a garden hose and closing a tap Let's say you have a garden hose connected to an ordinary water tap which is opened fully. If you pinch the end of the hose, water leaves the hose at a higher speed (and this can be useful while watering plants, to reach pots which are further away). However when a tap (with no hose connected) is opened only slightly, water flows out at a low speed, possibly even in drops.  The actions of pinching the end of a hose and of almost-closing an open tap seem similar, so why the difference in behaviour? Is this a dismissal by hit wicket? Consider the following situation,     A batsman's bat breaks while hitting the ball after which a piece of the   bat flies in the air, hits the stumps and the bails fall down.   Is this a dismissal by hit wicket? I'm looking for a library/method to parse an html file with more html specific features than generic xml parsing libraries.