Evaluate $$\sum_{n=0}^\infty (n+1)\left(\frac13\right)^n.$$   $$1 + 2\left(\frac{1}{3}\right) + 3\left(\frac{1}{3}\right)^2 + 4\left(\frac{1}{3}\right)^3 + 5\left(\frac{1}{3}\right)^4 +\cdots$$  Show using ‘Techniques of Convergence of Geometric Series’ only that it converges to 9/4?
How can I evaluate $$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$? I know the answer thanks to , but I'm more concerned with how I can derive that answer. It cites tests to prove that it is convergent, but my class has never learned these before. So I feel that there must be a simpler method.  In general, how can I evaluate $$\sum_{n=0}^\infty (n+1)x^n?$$
Minecraft Laucher has no Versions So this if the first time this has ever happened too me and I was wondering what it is. I just downloaded Minecraft and there is no area where i can change my version too 1.8.9/1.7.10 all i see when I do it is blank  Im just Wondering on how come even tho i hit the Repair button on the minecraft.msi download it dosnt fix it... If there is a way too fix this let me know cause my friend wants too play and I cant cause of this.. The system cannot find the file specified I have just downloaded WinEdt and written my first code, but it will not run because I keep getting the following error.  I am at a loss as to how to fix this.  Error launching Console Application PDFLaTeX ... Command Line: pdflatex.exe --interaction=errorstopmode --synctex=-1 &quot;practice.tex&quot; Startup Folder: C:\Users\shahand\WinEdt Team\WinEdt 8 The system cannot find the file specified. Challenges with Key with Gaps? I had a mistake with a trigger which resulted in errors in PostgreSQL 9.4.3 on x86_64-unknown-linux-gnu, compiled by gcc (Debian 4.9.2-10) 4.9.2, 64-bit.  Each time during the error, the primary key of SERIAL increased.  After fixing the bug, the table measurements is    measurement_id | measurement_size_in_bytes |             time               ----------------+---------------------------+-------------------------------               1 |                     77777 | 2015-07-14 18:29:56.858703+03               2 |                       888 | 2015-07-14 18:29:56.882552+03               3 |                       888 | 2015-07-14 18:30:15.505957+03               4 |                       888 | 2015-07-14 18:41:01.878106+03              39 |                     77777 | 2015-07-15 12:11:21.21391+03              40 |                     77777 | 2015-07-15 12:11:59.551973+03              41 |                     77777 | 2015-07-15 12:12:05.48982+03              42 |                     77777 | 2015-07-15 12:13:02.402053+03              43 |                     77777 | 2015-07-15 12:13:02.419412+03              44 |                       888 | 2015-07-15 12:13:02.434728+03          ...   where the amount of error statements has been 35 (= 39-4).  This jump in IDs may become a challenge later when I want to integrate this table with a partial index where order matters.   Table  CREATE TABLE measurements     (         measurement_id SERIAL PRIMARY KEY NOT NULL,         measurement_size_in_bytes INTEGER NOT NULL,         time TIMESTAMP WITH TIME ZONE NOT NULL DEFAULT CURRENT_TIMESTAMP,         CONSTRAINT no_duplicate_measurements UNIQUE (time)     );       What are the challenges of keys with gaps? I am wondering the following question: Let $X$ be a separable Banach space which is linearly isomorphic to a dual Banach space $Y^*$. Is there a Banach space $Z$ such that $X$ is lineraly isometric to the dual of $Z$: $X=Z^*$.  I think that the answer is no, but I do not have a counterexample. Since $L_1$ is not isometric to any dual Banach space, maybe one can find a dual Banach space which is isomorphic to $L_1$...  To finish, do that change anythink if I suppose $X$ to be almost linearly isometric to $Y^*$ ? By almost linearly isometric I mean that for every $\varepsilon &gt;0$ there exist $T$: $X \to Y$ a linear isomorphism satisfying $\|T\|  \|T^{-1}\| \leq 1+\varepsilon$. What is the "show desktop" keyboard shortcut? I just upgraded to 11.10 from 11.04. Previously the shortcut to show the desktop was Super +D (same as in Windows), But now it is not working. What is the new shortcut? The square roots of different primes are linearly independent over the field of rationals Previously, someone posted his/her proof of "every neighborhood is open" . I am focusing on the same statement, but it seems that my approach is different from his/hers. Could someone comment on my own proof? Here we go:  Let $X$ be a metric space and $p\in X$. Fix $r&gt;0$. Define a neighborhood around $p$ as $N_r (p)=\{q\in X| d(p, q)&lt;r\}$. Now, take $w = \frac{r}{2}$. Then, $N_w (p)\subset N_r(p)$. In other words, every point of $N_r(p)$ is interior. Therefore, $N_r (p)$ is open. Since $r$ is arbitrary, this proves every neighborhood is open.  Update:  Under my definition of "neighborhood," can I prove in this way instead: For all $m\in N_r(p)$, take $w=\frac{r−d(m,p)}{2}$. Then, $N_w(m)\subset N_r(p)$.   Update2  From the hints given in comments, I solve it:  Assume $m\in N_r(p)$ and $q\in N_w(m)$. Take $w=r-d(m, p)$. Then, $d(p, q)\leq d(p, m) + d(m, q)&lt; d(p, m) + r-d(m,p) =r$.   Thanks to @JoséCarlosSantos and @fleablood ! Why: a relative adverb, a conjunction ... or both? I am not familiar with the idea that an adverb can function as a conjunction at the same time. Here are a couple of sentences that are confusing me.     This is the reason why she left him.   ...and     He was transferred to New York, where he was promoted to a higher level.   The  why, where, when, and how seem to introduce a clause. Therefore, can they also be called conjunctions? If yes, would I be right in assuming that relative adverbs can function as conjunctions at the same time? PCA and the train/test split I have a dataset for which I have multiple sets of binary labels. For each set of labels, I train a classifier, evaluating it by cross-validation. I want to reduce dimensionality using principal component analysis (PCA). My question is:  Is it possible to do the PCA once for the whole dataset and then use the new dataset of lower dimensionality for cross-validation as described above? Or do I need to do a separate PCA for every training set (which would mean doing a separate PCA for every classifier and for every cross-validation fold)?  On one hand, the PCA does not make any use of the labels. On the other hand, it does use the test data to do the transformation, so I am afraid it could bias the results.  I should mention that in addition to saving me some work, doing the PCA once on the whole dataset would allow me to visualize the dataset for all label sets at once. If I have a different PCA for each label set, I would need to visualize each label set separately. Suppose $G$ has odd order, show the function $f:G\rightarrow G$ defined by $f(x)=x^2$ is injective. This proposition is easily provable if we assume $G$ is Abelian, but I don't know how to start this without the assumption of being Abelian. What is the distinction between a professional and amateur photographer? I understand that \bf and \it are now obsolete in LaTeX and that \textbf and \textit are proper, as they produce more sophisticated (in particular, cumulative and properly kerned) changes to font style. I have read the  and I understand the rules and their reasons, as well as the several other commands that are affected.  However, I find it convenient to use LaTeX for notetaking during lectures, and in that rushed environment, shortening a command by any number of keystrokes helps keep me from falling behind. \bf and its two-letter kin are still very useful to me for that reason, and once a presentation is finished I can go through and replace all appearances of \bf et al. with \textbf et al.  My question is this: is there a plan eventually to replace the short font style commands like \bf with the implementations of \textbf etc. some day, or should I expect \bf always to remain in existence but obsolete, for reasons of backward compatibility with original TeX? Original TeX has been greatly improved on in countless ways, but in the heat of transcription I sometimes miss its conciseness. As the question says, I need to edit my /proc/acpi/wakeup file, but when I open it with any text editor, it shows blank file. cat command, on the other hand, displays correct file contents in terminal. When I navigate to that file in file manager, it says it's size is 0 bytes. I'm using Xubuntu 15.04. I'm thoroughly confused. If the Standard Normal PDF is $$f(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}$$  and the CDF is  $$F(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-x^2/2}\mathrm{d}x\,,$$  how does this turn into an error function of $z$? In Excel, the window can be frozen in place in such a way that you can always see the name of the column even though you go way down below the 'fold' of the screen.  On SE, when you scroll, the whole page goes up (or down). This means the top navigation bar is scrolled off screen. I'm not sure if it would be a good idea or not to fix the top navigation to the top of the screen and let the rest of the content scroll.  What do you think about this, in terms of pros and cons?  [Update]  With the new Top navigation bar, kudos to the design team, I think this question becomes relevant again. I'm aware of the busload of downvotes, but still.  The top navigation bar doesn't occupy much space and has two notification areas:   Inbox Achievements   Having these two notifications visible at all times would be nice, IMO.  [Update]  I've just noticed that I haven't used the word "optional" in this discussion... My bad. In Excel, the Freeze function is controlled by the user. You can optionally Freeze part of the interface, and you can eliminate the Freeze at will. Desktop icons rearrange after using remote desktop connection I sometimes remote into my desktop PC (1900x1280) from my laptop (1280x900).  When I do, Windows (Vista) on the desktop PC rearranges all my desktop icons.  I could understand this if the RDP session's desktop size was too small to fit them in, but it's not.  I have about 12 icons, which do fit on the laptop's resolution (I've tried).  Why would Windows rearrange them all like this each time? I would like to count the number of unique selections from a column that contains multiple selections from a drop down list.  For example, column B3 contains,     Monday, Tuesday, Wednesday   The count function returns a value of 1 instead of 3 - is there any way to count the three days distinctly? Thank you!  Multiple selections from a drop down list was made possible using the VBA code from: How to serialize a JObject without the formatting? context menu parent? How do you prove the second law of thermodynamics from statistical mechanics?