Evaluating a sum: $\sum_{n=0}^{\infty} nx^{2n+1}$ My problem: I should calculate this sum: $\sum_{n=0}^{\infty} nx^{2n+1}$.   My solution: $\;$$ x \sum_{n=0}^{\infty} nx^{2n}$,   Substitution: $x^2 = y$   Then $\;$$ y^{1/2} \sum_{n=0}^{\infty} ny^{n}$, ...  Now I would focus just on $\;$$ \sum_{n=0}^{\infty} ny^{n} =  y\sum_{n=0}^{\infty} ny^{n-1}$$, $  Because $\;$$ \sum_{n=0}^{\infty} y^{n} = 1/(1-y)$  EDIT  So $   \sum_{n=0}^{\infty} ny^{n-1} = 1/ (1-y)^2$  Finally $y^{1/2}\sum_{n=0}^{\infty} ny^{n-1} = y^{1/2}/ (1-y)^2 = x/(1-x^2)^2$
How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$? How can I evaluate $$\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?$$
Counting the occurrences / frequency of array elements How Can I Expand A Tilde ~ As Part Of A Variable? When I open up a bash prompt and type:  $ set -o xtrace $ x='~/someDirectory' + x='~/someDirectory' $ echo $x + echo '~/someDirectory' ~/someDirectory   I was hoping that the 5th line above would have went + echo /home/myUsername/someDirectory.  Is there a way to do this?  In my original Bash script, the variable x is actually being populated from data from an input file, via a loop like this:  while IFS= read line do     params=($line)     echo ${params[0]} done &lt;"./someInputFile.txt"   Still, I'm getting a similar result, with the echo '~/someDirectory' instead of echo /home/myUsername/someDirectory. How to deal with hierarchical / nested data in machine learning When conducting multiple regression, when should you center your predictor variables & when should you standardize them? Using some matlab (I know it's cheating) I found that: $$k=0 =&gt; Result=1$$ $$k=1 =&gt; Result=1$$ $$k=2 =&gt; Result=2$$ $$k=3 =&gt; Result=5$$ $$k=4 =&gt; Result=15$$ $$k=5 =&gt; Result=52$$ $$k=6 =&gt; Result=203$$  This sequence is puzzling... When should you use WP_Query vs query_posts() vs get_posts()? It seems like half the tutorials in the  and around the blogosphere use  and half use . They all do similar things, so when should I use one over the others? Let's start with a definition:     Definition: Let $F$ be a field, $I$ a nonzero ideal in $F[x]$, and $g(x)$ an element of $F[x]$. Then, $I= \langle g(x)\rangle$ if and only if $g(x)$ is a nonzero polynomial of minimum degree in $I$.   Now it gets completely foggy what is meant with "minimum degree".I constantly come across the "minimal polynomial". This seems to be linked, but it seems strange that there is a division here.   Could someone give a pointer to what the second line actually means. Maybe some example or diagram (if possible) would help a lot here. https:// search results appearing on Google for purely http:// site I started weeding through my site's search results from Google today, using a site: search, to determine if there are any links that cause 404s and thus need redirecting. To my amazement I noticed numerous https:// results relating to various pages. My site doesn't have a SSL certificate, doesn't serve such pages, doesn't internally link to https:// pages, doesn't include any such files in its sitemap.xml and, for all of these, never has.  I decided to do a Google search for https://&lt;my site&gt; and found one site that incorrectly refers to the root of my site with a https:// prefix - I will try to contact them to get them to correct this.  I'm not sure however how Googlebot managed to index the non-root files as https://. I can't find any external links to them and surely, without certification, Googlebot should have stalled at the first request?  I've just added the following lines to the site's .htaccess (although the surfer still has to navigate through the browser's "This site is a security risk. Abandon hope all ye who enter here!" message(s) first to get there):  RewriteEngine On RewriteCond %{HTTPS} on RewriteRule ^(.*)$ http://www.&lt;my site&gt;.org/$1 [R=301,L]    replacing &lt;my site&gt; with my domain name.   My big question is this though - I would like to use the Google Webmaster Tools Remove URLs feature to remove the https:// pages from the index. Can I be guaranteed that this will only remove the https:// versions of each relevant page and not the valid http:// versions?  My thanks to anyone who can help me out with this particular question and the issue in general. "Give him a box that everyone knows what it contains."  Is this correct English? It sounds wrong to me. Double-blind review with four references in a row? I've read a lot of answers here about double-blind reviews and thesis or single articles.  My problem comes from having four self-references in my article. I feel that if I include them all, could be very easy to infer that this is the fifth one (It's a 4-year granted project and I have been collaborating in a number of software components inside a bigger architecture, now it's the time to use them all to interact with other applications) . But if I ommit some of them, I could end being accused of self-plagiarism. The other articles have been published in the proceedings of conferences that didn't require double-blind reviews, this is the first submission I make to a conference that asks for it (Maybe because of being CORE A) Also not worthless to mention that I'm going to send it as a short paper, so actually it'd be very helpful to cite rather than to repeat past texts.  What should I do? What is a good book to study linear algebra? I'm looking for a book to learn Algebra. The programme is the following. The units marked with a $\star$ are the ones I'm most interested in (in the sense I know nothing about) and those with a $\circ$ are those which I'm mildly comfortable with. The ones that aren't marked shouldn't be of importance. Any important topic inside a unite will be boldfaced.  U1: Vector Algebra. Points in the $n$-dimensional space. Vectors. Scalar product. Norm. Lines and planes. Vectorial product.  $\circ$ U2: Vector Spaces. Definition. Subspaces. Linear independence. Linear combination. Generating systems. Basis. Dimesion. Sum and intersection of subspaces. Direct sum. Spaces with inner products.  $\circ$  U3: Matrices and determinants. Matrix Spaces. Sum and product of matrices. Linear ecuations. Gauss-Jordan elimination. Range. Roché Frobenius Theorem. Determinants. Properties. Determinant of a product. Determinants and inverses.  $\star$ U4: Linear transformations. Definition. Nucleus and image. Monomorphisms, epimorphisms and isomorphisms. Composition of linear transformations. Inverse linear tranforms.  U5: Complex numbers and polynomials. Complex numbers. Operations. Binomial and trigonometric form. De Möivre's Theorem.  Solving equations. Polynomials. Degree. Operations. Roots. Remainder theorem. Factorial decomposition. FTA. Lagrange interpolation.  $\star$ U6: Linear transformations and matrices. Matrix of a linear transformation. Matrix of the composition. Matrix of the inverse. Base changes.   $\star$ U7: Eigen values and eigen vectors Eigen values and eigen vectors. Characteristc polynomial. Aplications. Invariant subspaces. Diagonalization.   To let you know, I own a copy of Apostol's Calculus $\mathrm I $ which has some of those topics, precisely:   Linear Spaces Linear Transformations and Matrices.   I also have a copy of Apostol's second book of Calc $\mathrm II$which continues with   Determinants Eigenvalues and eigenvectors Eigenvalues of operators in Euclidean spaces.   I was reccommended Linear Algebra by Armando Rojo and have Linear Algebra by , which seems quite a good text.   What do you reccomend? Can steam create an overview of all my achievements? I play several games on Steam and I love the achievement system. It would be great to get an overview of all the achievements ever gained in Steam games. Does Steam support this? I haven't found such an overview.  If such a feature does not exist, I would already be happy with an easy way to calculate the sum of all my achievements automatically.  Edit: I stumbled upon an interesting website called  today. Unfortunately it only works for XBox-Achievements. Does anyone know of a similar service for Steam? This would solve my question right away. Implementing my own solution based on the links posted in the answers didn't work because the APIs are not sufficient for my needs. I need some advice as I am struggling with the following combinatorics exercise.  Let $k$ be a given positive integer. Show that any non-negative integer $N$ can be written uniquely in the form $$N=\binom{x_k}{k}+\binom{x_{k-1}}{k-1}+\dots+\binom{x_1}{1},$$ where $0 \leq x_1 &lt; \dots &lt; x_{k-1} &lt; x_k$.  The exercise is accompanied by this advice:  Let $x$ be such that $\binom{x}{k} \leq N &lt; \binom{x+1}{k}$. Then any possible representation has $x_k = x$. Now use induction and the fact that $N-\binom{x}{k} &lt; \binom{x}{k-1}$ to show the existence and uniqueness of the representation.  I am really stuck and thus grateful for the slightest hint on how to solve it. MVC Spring Implementation - error: java.lang.NullPointerException What is the proper way of creating installation media from Ubuntu iso? Ephemeral error message: "cryptsetup: unknown fstype, bad password or options?" Removing repeated page numbers from table of contents Hi I'm trying to create a document in LaTeX with a few small subsections on the same page, is there any way to remove the repeated page numbers and keep the numbers when I move onto a new page? Proof that the nth root of a rational s/t is irrational unless s and t are perfect nth powers I'm trying to prove that $\sqrt[n]{\frac{s}{t}}$ is irrational unless both s and t are perfect nth powers. I have found plenty of proofs for nth root of an integer but cannot find anything for rationals. Also trying to work up from the proofs I have found is rather difficult.  My lecturer wants me to prove this using uniqueness of prime factorisation but I also have no idea where to start with that.   Any help is greatly appreciated. Parallel execution of bash script I have a table URL_Experiment in my database (mySQL database). I have 2 million URL links in this table.  For each URL, I am checking if some particular text is present in the URL and updating the information back in my URL_Experiment table.    Now, I need to make the script run fast. The easy solution that comes to my mind is:  split the URL_Experiment into 10 tables (each with 200,000 rows) and run the script 10 times simultaneously. This way, I can ensure that the updates happen properly. But for this, I need to save 10 scripts, each accessing the correct table (I need to rename the tables from 1 to 10).   Though the above seems a pretty neat solution, I am looking for a more sophisticated solution. Something which will execute the same script simultaneously (in parallel) as 10 processes. My concern is the execution time. I do not want the resources to be wasted when they can be utilized. What happens if you call erase() on a map element while iterating from begin to end?