[enter image description here][1]  $$\sum_{k=1}^{\infty} k[(1-p)^k +p^k(1 -p)] =(1-p)p \sum_{k=1}^{\infty} [k(1-p)^{k-1} +\sum_{k=1}^{\infty} kp^{k-1}]$$  $$=(1-p)p + [\frac{1}{p^2} +\frac{1}{(1-p)^2}] =\frac{1-2p+2p^2}{p(1-p)}$$   Can someone please let me know how the 2 summed series simplified to $\frac{1}{p^2}$ and $\frac{1}{p-1}^2$   I suspect it to be a geometric series and I seem to have $\frac{1}{p}-\frac{1}{1-p}$
Find a closed formula for $\sum_{n=1}^\infty nx^{n-1}$  I am trying to use the derivative of generalized binomial theorem, $\frac{d}{dx}[(x+1)^r=\sum_{n=0}^\infty \binom{r}{n}x^n] =r(x+1)^{r-1}=\sum_{n=1}^\infty \binom{r}{n}nx^{n-1}$  However, I am not sure how to "get rid" of the binomial term.
"The news is good." Why? For days I tried several ways to install Windows 10 . With Windows Update I have always an error and with the media creation tool microsoft I have this message: " The installation failed in SAFE_OS phase with an error when BOOT operator " with this code : 0xC1900101 - 0x20017 Having tried everything I do not know what to do ... Can you help me please ? Thank you  PS: I migrated my OS with the software "Paragon migrate OS to SSD" from my hard disk to my SSD. Maybe I have this errors because I did this.... Missing abstract with amsart class I'm not very good at LaTeX but I've been using it to write my dissertation. Have just come to the end and trying to format everything correctly. Whenever I try to put my Abstract paragraph into the form \begin{abstract} (writing here) \end{abstract} and click play the whole abstract section just disappears. What am I doing wrong?  \documentclass[12pt]{amsart}  \begin{document}  \begin{abstract} (my abstract here) \end{abstract} \end{document} How to pass parameters in batch file and zip the file using 7z.exe? I need to create a batch file as per below requirements  1) pass path1 &amp; path2 while running the batch file  exl: testrun.bat E:\ERP\test1.txt E:\ERP\header.txt  2) Need to store these command line paths values in batch file  exl: var1=E:\ERP\test1.txt var2=E:\ERP\header.txt  3) zip the path1 using PKZIPW.exe or 7z.exe and create new o/p zip file  exl: 7z.exe u -tzip E:\Erp\Test1.zip var1  4) copy path2 in newly created o/p file  exl: copy /b var2+test1.zip E:\Erp\Final_output.zip Rule of thumb for number of bootstrap samples How do I get the value of text input field using JavaScript? Using Unicode Zero Width Spaces allows all space comments Filter "Hot Network Questions" by excluding sites Can you retrain the Magic Initiate spell using the Spellcasting feature? got me thinking about the mechanics of the Magic Initiate feat when you have class levels in the class into which you were inducted.    Among the classes you can pick with this feat are the Bard, Sorcerer, and Warlock, which know a limited number of spells and all have the same statement under Spells Known.     Additionally, when you gain a level in this class, you can choose one of the [class] spells you know and replace it with another spell from the [class] spell list , which must also be of a level for which you have spell slots.   The term "learning" is used both to describe new spells acquired on level-up and the spell acquired through the feat. This would seem to indicate your Magic Initiate spell counts among the spells you know, and is thus eligible for retraining if you learned it as a spell of your class.  An odd change in wording occurs during the Magic Initiate feat description (emphasis mine):     In addition, choose one 1st-level spell from the same list. You learn that spell and can cast it at its lowest level.   This generalization in describing the learned spell's level seems to imply that there exists a scenario where it would not be appropriate to restrict the spell to being cast as a 1st-level spell.  Could one of the charisma casters retrain Magic Initiate into a higher level spell, which they could then cast once per long rest without using a slot?  This would turn an otherwise unexciting feat into, effectively, a Mystic Arcanum for Bards and Sorcerers. This seems too powerful to be intentional,  but the feat description appears to have gone out of its way to permit it. When we lie down, the "back part of the body" is in contact with the bed. It includes the heels, the calves, the backside, the back, and the back of the neck, and the back of the head etc.  How do we refer to the entire "back part" of the body? I have had several business loans and understand that this is not by any means typical.  I noticed that for a year my payments were not decreasing the principal on the loan. When I asked the bank why, I found out that at some point my payments were getting applied to principal only. Now I am paying interest off that was not paid for 6 months or so.  My banker states that he can change all the interest payments to principal payments. I am confused to why this would be. If I make all my principal payments first then would the bank lose a substantial amount of interest?  This has happened on 2 of the 3 loans I have looked at so far.  If they are willing to change all the payments to principal only, would this be beneficial to paying less interest ultimately? I assume there would be an interest account accruing somewhere that would need to be paid off at the end of the loan.  I understand that typically you have a loan balance that would include interest and principal. My balance owed did not change when I continued to make same payment? It gets applied to interest. The balance on the statement must not be a true balance. There must be interest owed above the balance showed but why would they do this to start with? They applied full payments to principle only first in order for interest to get to be more than several payments What and how does Darth Vader see? Are there any sources that talk about  about Darth Vader's visual acuity?  His life-supporting armour is quite advanced but does it provide any useful augmentation to his visual abilities? $\prod_{k=0}^{\infty} \cos(x \cdot 2^{-k}).$ Task is to find $$\prod_{k=0}^{\infty} \cos(x \cdot 2^{-k}).$$  I tried to make it with double-angle formula: $\prod_{k=0}^{\infty} \cos(x \cdot 2^{-k}) = \frac{\prod_{k=0}^\infty \sin(x2^{1-k})}{2^\infty \cdot \prod_{k=0}^\infty \sin(x \cdot 2^{-k})} $ I'm sad to admit, that I'm stuck with that one. So I would appreciate any help, such as pointing the right direction. How long are the life phases of a dragon? I'm new to field extensions and I can neither see why or how to prove this statement.  If I'm not mistaken, to construct a splitting field, you construct rupture fields one after the other, using that the $8$ element field is the splitting field of $X^8-X$ over $\mathbb F_2$, since $2$ is the only prime divisor of 8.  Over $\mathbb F_2$ $X^8-X$ has two roots : $0$ and $1$. Therefore $X^8-X=X(X-1)(1+X+...+X^6)$. The last factor, call it $P$, has no roots. So let $\alpha$ be a root in some extension $\mathbb F_2(\alpha)$ of $\mathbb F_2$.  We now have $P=(X-\alpha)(X^5+\alpha X^4+...+\alpha^4X+\alpha^5)$.  I suspect $\alpha$ might not be the way to go and if so, my second plan would be to reduce $P$. I don't know how to do this but assuming it to be done, hence having $P=P_1...P_n$, I could start using $P_1$'s rupture field, and then my best guess is that if $\mathbb F_2[X]/(P_1)$ is not sufficient, I re-reduce $P$ over this new field and consider the rupture field of one of the factors, and so on.  Now assuming I wasn't given the statement itself, is it elementary to construct the $8$ element field using $\mathbb F_2$ ? How to change boot priority? I have a laptop with Ubuntu installed as the only OS. I also have a USB stick with a Windows 7 installation on it. I want to install Windows from the USB stick on my laptop but when I try to restart the laptop to get access to BIOS, I can't seem to find the command for it, its like it skips everything and just says Ubuntu and then takes me to the login screen.  How do I change the boot priority to USB in Ubuntu? Prove that if a function's average is nondecreasing then the function is nondecreasing. Let $\omega:\mathbb{R}_+\to\mathbb{R}_+$ such that $\omega(0) = 0$ and $\xi(s) = \frac{1}{s}\int\limits_{0}^s \omega(t)dt$ is nondecreasing, i.e., $\forall s,s': s\leq s'$, $\xi(s)\leq \xi(s')$. Show that $\omega(t)$ is nondecreasing a.e., i.e. find a function $w(t)$ such that $w$ is nondecreasing and $\omega = w$ a.e.  I was able to show the reverse implication easily, that if $\omega$ is nondecreasing then $\xi$ is nondecreasing. This way seems more difficult. Correlations between continuous and categorical (nominal) variables I would like to find the correlation between a continuous (dependent variable) and a categorical (nominal: gender, independent variable) variable. Continuous data is not normally distributed. Before, I had computed it using the Spearman's $\rho$. However, I have been told that it is not right.  While searching on the internet, I found that the boxplot can provide an idea about how much they are associated; however, I was looking for a quantified value such as Pearson's product moment coefficient or Spearman's $\rho$. Can you please help me on how to do this? Or, inform on which method would be appropriate?  Would Point Biserial Coefficient be the right option? OpenSSL remains vulnerable after update to 14.04 How to ignore accent in SQLite query (Android)