I was trying to create closed formulas for certain iterated sequences, and came across this sum: $$A := \sum_{k=0}^n q^{2^k} = q+q^2+q^4+\dots+q^{2^n}, \quad q\in\Bbb{R},n\in\Bbb{N}^+$$ I'm trying to find a closed formula for this expression, or a formula where you can obtain the value of the sum by using much fewer arithmetic operations.  I know that the sum  $$S := \sum_{k=0}^n q^k = 1+q+q^2+\dots+q^n$$  has a closed form, since  $$\begin{align} S &amp;= 1+q+q^2+\dots+q^n \\ qS &amp;= q+q^2+q^3+\dots+q^{n+1} \\ qS-S &amp;= q^{n+1}-1 \\ (q-1)S &amp;= q^{n+1}-1 \\ S &amp;= \begin{cases} \frac{q^n-1}{q-1} \quad \text{ if } q \ne 1\\[2ex] n+1 \quad \text{ if } q = 1\end{cases} \end{align}$$  But when trying to use a similar method to determine the value of $A$, we just get a tautology:  $$\begin{align}A = q+q^2+q^4+\dots+q^{2^n}\end{align}$$  Multiplying with $q^c$ won't result in any sort of simplification, so it makes the most sense, to square both sides:  $$\begin{align} A^2 &amp;= \begin{matrix}q^2+q^4+q^8+\dots+q^{2^{n+1}} + q(q^2+q^3+\dots+q^{2^n}) + \\ + q^2(q+q^3+\dots+q^{2^n})+\dots+q^{2^n}(1+q+\dots+q^{2^{n-1}})\end{matrix} \\ A^2 &amp;= A-q+q^{2^{n+1}}+q(A-q)+q^2(A-q^2)+\dots+q^{2^n}(A-q^{2^n}) \\ A^2 &amp;= A-q+q^{2^{n+1}}+qA-q^2+q^2A-q^4+\dots+q^{2^n}A-q^{2^{n+1}} \\ A^2 &amp;= A-q+q^{2^{n+1}}+A(q+q^2+\dots+q^{2^n})-(q^2+q^4+\dots+q^{2^{n+1}}) \\ A^2 &amp;= A-q+q^{2^{n+1}}+A^2-(A-q+q^{2^{n+1}}) \\ A^2 &amp;= A^2 \dots \end{align}$$  I also tried writing up $A$ in the $q$-number system, such that:  $$A = \left(\stackrel{q^{2^n}}{1}\ \underbrace{00\dots 0}_{n-1}\ \stackrel{q^{2^{n-1}}}{1}\ \underbrace{00\dots 0}_{n-2}\ \stackrel{q^{2^{n-2}}}{1} \ \dots \ \stackrel{q^{2^2}}{1}\underbrace{\stackrel{q^3}{0}}_{1}\stackrel{q^{2^1}}{1}\stackrel{q^{2^0}}{1}\stackrel{q^0}{0}\right)_{q}$$  And tried to use formulas to fill up the $0$'s, and relate $A$ to $(11\dots 1)_{q}$, but I just got even more complicated equations that would be more difficult to write up in a closed formula than $A$.  I'm doubtful of there being such a formula, but maybe we can actually obtain one using a different approach.  WolframAlpha also couldn't simplify it, however, that still doesn't mean that such a closed formula doesn't exist.  I'm aware of the trick that  $$A = q+q^2+q^4+\dots+q^{2^n} = q(1+q(1+q^2(1+q^4(1+\dots(1+q^{2^{n-1}})\dots),$$  which involves fewer arithmetic operations to calculate, however I'm only satisfied with a closed formula if it doesn't include dots and/or sum/product/etc. notations. If you could help me find one or show that it doesn't exist using our basic arithmetic notations, that'd be highly appreciated! Thank you in advance.
I am wondering if there exists any formula for the following power series :  $$S = r + r^2 + r^4 + r^8 + r^{16} + r^{32} + ...... + r^{2^k}$$  Is there any way to calculate the sum of above series (if $k$ is given) ?
Car accident, put in park or neutral? How to redirect complete outgoing phone's traffic to a proxy server? How can I attach a file to a question? One of the answers in a reading exercise in my class today was:   100,000,000,000,000,000,000   ... which was the value of the highest denomination note ever issued. It was a 100,000,000,000,000,000,000 Pengo note, issued by Hungary in 1946. The students in my class wanted to know how they could say that last sentence in words in English. So I told them that I didn't know. They pointed out to me that if they wanted to read this bit of writing to someone they'd want to be able to use the words. Fair point.  Can anyone help?  [In case you're interested, a 100,000,000,000,000,000,000 Pengo note was worth about $0.20!] What term refers to those who adhere to scientism? Is there any differences between      Try to do   And     Try doing   Thank you in advence. I have looked it up and the resources all talk about schengen visa, travelling with eu national family member etc. so I would like to get a more assuring answer about my situation:   I am a non-EU national with an Italian residence permit. My father is an Italian citizen. I will be travelling to Sofia and Bucharest next month with my girlfriend who is a British citizen. Do I need to get a visa to enter these two cities? What is IEEE 802.3 Ethernet's communication type: Synchronous or Asynchronous? In the IEEE 802.3 frame structure there is a 7 byte preamble section for synchronization. If it is really for synchronization does that mean Ethernet is synchronous and not asynchronous?    Ethernet structure for reference: Generic constraints, where T : struct and where T : class Override iTunes "media" keys (play, pause, etc) for Spotify? How to zoom in/out an UIImage object when user pinches screen? Why are light rays able to cross each other? See the image first:   Why are light rays able to cross each other? Air isn't able to. I quite often add a question as a favourite when there is something of interest to me in it or its answers. Looking back through my favourites, I cannot remember why I added some of them as a favourite.  What I'd like to be able to do is to add a note/comment when I add a favourite as a reminder as to why I've added it. This will show on my favourites page, but nowhere else - and possibly to no one else.  In addition, it would be good to be able to favourite an answer to a question, thereby adding a favourite to both the question and a particular answer with something of note in it. This would only show as a single favourite, but when clicking its link would go to the favourited answer.  Possible? Or is there another way of doing this? I've already got too many favourites to remember why I added a lot of them. tcolorbox Spanning Two Columns in {paracol} Environment The paracol environment allows figures to span two columns by using an asterisk in the figure's environment declaration: \begin{figure*} ... \end{figure*}.  I'm not familiar with the guts of why that works, and I was curious to know if there's something analogous for an environment derived from tcolorbox (like \newtcolorbox{NewBox}[1][!htbp]{...float=#1}). That is, in this example, can I get a NewBox that spans both paracol columns the way I wrongly assumed \begin{NewBox*} ... \end{NewBox*} would.  PS The tag below is "parcolumns" because I don't have a high enough reputation to create the tag 'paracol' (in reference to this CTAN package: ) Is there a rule of thumb for objective questions asked out of curiosity? How can I recursively delete all files of a specific extension in the current directory? How do I safely delete all files with a specific extension (e.g. .bak) from current directory and all subfolders using one command-line? Simply, I'm afraid to use rm since I used it wrong once and now I need advice. How to compute $\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}dt$? Calculating with Mathematica, one can have $$\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}\,\mathrm dt=\frac{\pi}{4}.$$   How can I get this formula by hand? Is there any simpler idea than using $u = \sin t$?     Is there a simple way to calculate $$ \int_0^{\pi/2}\frac{\sin^n t}{\sin^n t+\cos^n t}\,\mathrm dt $$ for $n&gt;3$? Could anyone come up with a reference for this exercise? I want to hit enemy's axe to break it. How this is properly played in D&amp;D 5e? And what about shield or armor? Can I attack these items to break it? Why not use tables for layout in HTML? It seems to be the  that tables should not be used for layout in HTML.   Why?  I have never (or rarely to be honest) seen good arguments for this. The usual answers are:   It's good to But this is a fallacious argument; . I guess it's true that using the table element for layout has little to do with tabular data. So what? Does my boss care? Do my users care?Perhaps me or my fellow developers who have to maintain a web page care... Is a table less maintainable? I think using a table is  than using divs and CSS.By the way... why is using a div or a span good separation of content from layout and a table not? Getting a good layout with only divs often requires a lot of nested divs. Readability of the codeI think it's the other way around. Most people understand HTML, few understand CSS. It's better for SEO not to use tablesWhy? Can anybody show some evidence that it is? Or a statement from Google that tables are discouraged from an SEO perspective? Tables are slower.An extra tbody element has to be inserted. This is peanuts for modern web browsers. Show me some benchmarks where the use of a table significantly slows down a page. A layout overhaul is easier without tables, see .Most web sites that need an upgrade need new content (HTML) as well. Scenarios where a new version of a web site only needs a new CSS file are not very likely. Zen Garden is a nice web site, but a bit theoretical. Not to mention its  of CSS.   I am really interested in good arguments to use divs + CSS instead of tables. I have built my house in Minecraft PE, and I have a ton of fences and sticks, but I need to know how big of a perimeter I need to let me sleep at night?