How does one deduce the approximation of $\sqrt{7}$ to be $\frac{10837}{4096}$ by taking $x = \frac{1}{64}$ in the expansion of $\sqrt{1-x}$?  How should you approach such a question? I assume the first step would be to expand $\sqrt{1-x}$ which can only be done through binomial theorem (afaik).   That gives $\sqrt{1-x} = 1 - \frac{1}{2}(x) - \frac{1}{8}(x^2)$ + ... and so on.  How do you continue? I can't seem to figure out how taking $x = \frac{1}{64}$ accomplishes anything.
I am trying to solve a 2 part question, the first part is to expand $\sqrt{1-x}$ up to and including the term $x^2$ which I did.  This gives me $1-(0.5)x-(0.125)x^2 + ...$  However, the 2nd part of the question says that by taking $x=(1/64)$ in the expansion of part (i) -> which refers to the one above, deduce that $\sqrt 7 = 10837/4096$. How do i do this part?  EDIT (After Hint Given By SimpliFire):  $$ \sqrt{1-\frac1{64}}=\frac{\sqrt{63}}8=\frac38\sqrt7 $$ $$ =\frac83\sqrt{1-\frac1{64}}\;=\sqrt7 $$ $$ =\frac83\left(1-\frac12\left(\frac1{64}\right)-\frac18\left(\frac1{64}\right)^2\right)\; $$ $$ =\frac83\left(1-\frac1{128}-\frac1{32768}\right)\; $$ $$ =\frac83\left(\frac{32768}{32768}-\frac{256}{32768}-\frac1{32768}\right)\; $$ $$ =\frac83\left(\frac{32511}{32768}\right)\; $$ $$ =\frac{260088}{98304} $$ $$ \frac{10837}{4096}\;\approx\sqrt7 $$  I'm not sure if I can present it like this, if it is okay i will post it as my answer :X
How can I know if a branch has been already merged into master? Given an InputStream called in which contains audio data in a compressed format (such as MP3 or OGG), I wish to create a byte array containing a WAV conversion of the input data. Unfortunately, if you try to do this, JavaSound hands you the following error:  java.io.IOException: stream length not specified   I managed to get it to work by writing the wav to a temporary file, then reading it back in, as shown below:  AudioInputStream source = AudioSystem.getAudioInputStream(new BufferedInputStream(in, 1024)); AudioInputStream pcm = AudioSystem.getAudioInputStream(AudioFormat.Encoding.PCM_SIGNED, source); AudioInputStream ulaw = AudioSystem.getAudioInputStream(AudioFormat.Encoding.ULAW, pcm); File tempFile = File.createTempFile("wav", "tmp"); AudioSystem.write(ulaw, AudioFileFormat.Type.WAVE, tempFile); // The fileToByteArray() method reads the file // into a byte array; omitted for brevity byte[] bytes = fileToByteArray(tempFile); tempFile.delete(); return bytes;   This is obviously less desirable. Is there a better way? What Canon lens would one use for Sartorialist-like street photography? You probably know Scott Schuman and his blog (). What kind of lens (that is not expensive) would you recommend for Canon 550D to achieve similar effect (in terms of DoF etc.)? I'm a newbie so I'm not sure how to properly interpret those photos' EXIF info. My guess is, that a similar effect can be achieved using .   PS:   Update: Thank you guys for all the answers, it's much more clear for me right now. I think I'll try the Canon 50mm f/1.4 lens. crossfade between 2 videos using ffmpeg Illustrating op amp feedback without control theory We are teaching op amps in a high school class, before calculus.  So we can't use control theory to teach how op amps respond.  All the same, I would like to have an intuitive explanation for how a feedback circuit works.  Take the negative feedback, for instance.  Is there a clear way to show how an initial difference of delta between V+ and V- leads to a very large (G(V+ - V-)) difference in the output, which then leads to ...  I'd like to be able to take that argument and show how the voltage of the output converges in addition to the standard virtual short argument.  Can anyone explain it clearly? list transparency across non consecutive slides Suppose $X$ and $Y$ are independent positive random variables with probability density functions $f_X$ and $f_Y$ respectively. Show that $Z=X/Y$ is absolutely continuous and find its probability density function.   $\textbf{My Thought:}$ In order to show that $Z$ is absolutely continuous, I need to show that it has a commulative distribution function. Then I can also differentiate the latter to obtain the probability density function of $Z$. I have the following calculation \begin{align*} P(X/Y\leq z)&amp;=P(X\ge zY,Y&lt;0)+P(X\le zY,Y&gt;0)\\ &amp;=\int_{-\infty}^{0}\left(\int_{yz}^{\infty}f_{X}(x)dx\right)f_{Y}(y)dy+\int_{0}^{\infty}\left(\int_{-\infty}^{yz}f_{X}(x)dx\right)f_{Y}(y)dy\\ &amp;=\int_{0}^{\infty}\left(\int_{0}^{yz}f_{X}(x)dx\right)f_{Y}(y)dy. \end{align*} Also, if $z&lt;0$, then we have that $F_Z(z)=0,$ since $X,Y$ are positive random variables. Therefore the random variable $Z$ is continuous, and we can differentiate it to obtain the probability distribution function of $Z$. We have that $f_Z(z)=0$ if $z\leq 0,$ and otherwise we have $$f_Z(z)=\int_{0}^{\infty}yf_{X}(yz)f_Y(y)dy.$$    Is my reasoning above correct?  Any feedback is much appreciated.  Thank you for your time. Let $R$ be a ring and let $J$ be an ideal of $R$. Assume $J$ contains   a unit of $R$. Prove $J=R$, rigorously.   Again I feel here as if the statement is intuitive so I am not able to make it rigorous. If you could show all steps in the proof that would be nice. Evaluation of the limit $\lim\limits_{n \to \infty } \frac1{\sqrt n}\left(1 + \frac1{\sqrt 2 }+\frac1{\sqrt 3 }+\cdots+\frac1{\sqrt n } \right)$ Evaluate the limit : $$\lim_{n \to \infty } {1 \over {\sqrt n }}\left( {1 + {1 \over {\sqrt 2 }} + {1 \over {\sqrt 3 }} + \cdots + {1 \over {\sqrt n }}} \right)$$  I can use the sandwich principle, certain convergence criteria, Cesaro mean theorem, limit arithmetic.. things around this area.  Any help would be greatly appreciated, thanks!  Sorry for not elaborating more at the beginning, rookie first-post mistake I suppose. :) Finding the perimeter of a right triangle with area given The expected value of $1/(1+X)$ where $X$ has geometric distribution Let $X$ be a random variable with a geometric distribution with parameter $p$.       Calculate $\Bbb E[1/(1+X)]$   I've looked at answers online but still do not understand how to solve this.   Specifically, I do not understand how to to get from $$\frac{p}{1-p} \sum_{x=0}^\infty (1/(1+x))((1-p)^x+1)$$  to  $-p\ln(p)  /(1-p)$. C++ single template specialisation with multiple template parameters TikZ rectangular node with different rounded corners Print  only? How do I print the indicated div (without manually disabling all other content on the page)?  I want to avoid a new preview dialog, so creating a new window with this content is not useful.  The page contains a couple of tables, one of them contains the div I want to print - the table is styled with visual styles for the web, that should not show in print. Let me add or change edit summary during edit window (grace period) Clerical error in UK visa application, but form already submitted - what should I do? I'm from the Philippines, and I plan on travelling to the UK with my sister to visit a relative.  I have just submitted my visa application and have already paid online, when I noticed that I've made a clerical error. In the 'visit or stay' portion of section "Do you have any friends or relatives in the UK?", I clicked the 'no' box, instead of 'yes'.  My sister's application is correct. She selected 'yes'. Our appointment is due next month. I am planning to email them about this, and also inform them on the day of application submission.  Is this the right path to take? There are 100 ropes in a bag. In each step, two rope ends are picked at random, tied together and put back into a bag. The process is repeated until there are no free ends. What is the expected number of loops at the end of the process? How to prepare data for ListVectorPlot[]? I really like the 'ring chart' or 'wheel chart' given as an example on page 88 of the PGF manual. It is like a pie chart, but with a label in the centre. Unfortunately, the manual doesn't give the source for that chart. Does anyone have an example? Thanks. Render scene with background jpg in python Fairly new to python and blender, so apologies if this is super basic. I've searched around and haven't found an answer.  I'd like to render some procedurally generated objects with a background image. In case it makes a difference, I'm using:    I don't need any interaction between the generated object and the background image so I've left a skeleton script I'm using to render below  import os DIR ="/home/path/to/folder" bg_file = os.path.join(DIR, "myimage.jpg") bg_img = bpy.data.images.load(bg_file) bpy.data.scenes['Scene'].render.filepath = os.path.join(DIR, "image") bpy.context.scene.camera = bpy.data.objects['Camera'] bpy.ops.render.render(write_still=True)   I managed to find some scripts that seem to be trying to do what I want online, but none that I could get to work. e.g.  import bpy filepath = "/home/path/to/folder" img = bpy.data.images.load(filepath)  for area in bpy.context.screen.areas:     if area.type == 'VIEW_3D':         space_data = area.spaces.active         bg = space_data.background_images.new()         bg.image = img         break  cam = bpy.data.cameras.new("Camera") cam_ob = bpy.data.objects.new("Camera", cam) bpy.context.scene.camera = cam_ob bpy.context.scene.objects.link(cam_ob) bpy.context.scene.objects.active = bpy.context.scene.objects["Camera"]   Any thoughts/tips would be greatly appreciated!