I have the following series:  $$ \sum_{n=1}^{\infty}n\theta^n, 0 \lt \theta \lt 1$$  I've seen lots of resources saying this is convergent. And using convergence tests to prove that.   Question:     But is there any formula which gives the summation in closed form?
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 $$\sum_{n=0}^\infty (n+1)x^n?$$
Why is the root directory denoted by a / sign? I have done some research about this on Google, but the results were cloudy. Why is the / sign used to denote the root directory. Are there any solid reasons behind it? Prove that if every node in a simple graph $G$ has degree $3$ or higher, then $G$ contains a cycle with a chord. Continuous and bounded imply uniform continuity? adding labels to points in ListPlot Show that $\frac{X-\mu}{\sigma}\sim N(0,1)$ using moment functions Let $X\sim N(\mu,\sigma^2)$. Show that $Z=\frac{X-\mu}{\sigma}\sim N(0,1)$ using moment generating functions. \begin{align*} M_Z(t)&amp;=M_{\frac{X-\mu}{\sigma}}(t)\\ &amp;=M_{X-\mu}\left(\frac t\sigma\right)\\ &amp;=e^{-\mu t}M_X\left(\frac t\sigma\right)\\ &amp;=e^{-\mu t}\cdot e^{\frac{t\mu}{\sigma}+\frac{t^2}{2}}\\ &amp;=e^{-\mu t+\frac{t\mu}{\sigma}+\frac{t^2}{2}} \end{align*} I'm not sure how to conclude that $Z\sim N(0,1)$. On webmail every user gets error "ERROR: Message not sent. Server replied: 354" while sending mails.   But mails are sent properly and reaches properly.  Can you tell me where is problem, and why it is occurred?  UPDATE: I found in some resources that this is not an error but standard information message. On the other side, it's annoying. Can you tell me if it is a true, and if it is how can I avoid it? How to find web hosting that meets my requirements? This is a "catch-all" question designed to serve as an answer for all questions about choosing web hosting.  Pro Webmasters no longer accepts new questions about how to choose hosting.  All future questions pertaining to finding web hosting should be closed as a duplicate of this question. For more information about this policy please see .    How to find web hosting that meets my requirements?  What we're looking for in answers to this question are the basics about web hosting:   What is web hosting? What is the difference between shared, VPS, and dedicated hosting? How does a content delivery network relate to web hosting? Anything else you feel is helpful in finding a web host.   What we do not want is:   Endorsements or recommendations for specific web hosts We do not want your experience or other subjective information (just the facts please) Redirect STDERR and STDOUT to different variables without temporary files \footnote breaks @-constructs in \index What is the origin of elements heavier than iron? In all the discussions about how the heavy elements in the universe are forged in the guts of stars and especially during a star's death, I usually hear that once the star begins fusing lighter atoms to produce iron (Fe) that's the end of the star's life and the whole system collapses onto itself; and based on how massive the star was initially, it has different outcome - like a white dwarf, a neutron star or a black hole. I have rarely heard a detailed explanation of how the elements heavier than iron are produced. I would appreciate a convincing explanation of this process. Prove that for any positive integer $n$,       $$n^{1/n}  &lt; 1 + \sqrt{\frac{2}{n}}.$$   This due to Victor Linis, Eureka, Vol. 2, No. 2, February 1976, p. 29.   Hint:      Use the binomial theorem. Strange result when removing item from a list while iterating over it Footnotes in tables? How do you install Google Chrome on Ubuntu? Is there a simple way to explain how Kepler's third law follows from the inverse square law that of gravity  (and laws of motion)  For example for Kepler's second law we can say it's because Gravity is a central force  What is it about the law of gravity which ultimately gives rise to Kepler's third law? How can I delete an email address from Outlook's email history? I'm having problems with an exercise from Apostol's Introduction to Analytic Number Theory.     Given $x$ and $y$, let $m=ax+by$, $n=cx+dy$, where $ad-bc= \pm 1$. Prove that $(m,n)=(x,y)$.   I've tried to give a proof, but I suspect it's wrong (or at least not very good). I would be very thankful for any hints/help/advice!  My proof:     We observe that since $ad-bc= \pm 1$, $ad=bc \pm 1$, and $(ad,bc)=1$.   Now, $(a,b) \mid m$ and $(c,d) \mid n$ but      $$(a,b) = ((d,1)a,(c,1)b)=(ad,a,bc,b)=(ad,bc,a,b)=((ad,bc),a,b)=(1,a,b)=1.$$      Similarly, we determine $(c,d)=1$. So, $1=(a,b) \mid m$ and $1=(c,d) &gt; \mid n$. But $(x,y)$ also divide $m$ and $n$. Since $(x,y) \geq (a,b)=(c,d)=1$, this implies that $(x,y)=m,n$. Hence   $(m,n)=((x,y),(x,y))=(x,y)$. How does PK know better than Jaggu? I'm using Ubuntu 14.04 but since some weeks I'm stuck with the old 13.10 lockscreen design for some reason.  I already tried compizconfig-settings-manager but it doesn't offer me any option for changing the lockscreen design and even resetting Compiz and Unity with   dconf reset -f /org/compiz/   didn't help.  Are there any other ways to get the new lockscreen back? Avoiding getting suspended for voting irregularities? Many times when I read threads, I keep on upvoting for the reason being that if someone has put an effort to write a big answer, s/he/it should get some rewards for it.  I recently saw someone getting suspended for voting irregularities.   So, I was wondering if I use the total amount of votes allowed per day, daily, will I be suspended?