I would like to compute the value of the following sum:     $\sum_{k\leq m} (-1)^k\binom{n}{k}$    for an integer m and a non negative integer n.  Progress  I know $\sum_{k=0}^n (-1)^k\binom{n}{k} = 0$ and symmetry of binomial-coefficients can be used to simplify some cases but beyond that i'm stuck.
It is easily checked that $\displaystyle\sum_{i\ =\ 0}^{n}\left(\, -1\,\right)^{i} \binom{n}{i} = 0$, for example by appealing to the binomial theorem.  I'm trying to figure out what happens with the truncated sum $\displaystyle\sum_{i\ =\ 0}^{D}\left(\, -1\,\right)^{i}\binom{n}{i}$.  How far away from $0$ can this get, as a function of $D$ ?.  I'm mostly interested in the case of when $D \ll n$, such as $D \sim \,\sqrt{\,n\,}\,$.  Thanks !
Links to the Wayback Machine are broken in chat transcript is an example of the bug. It looks like this in regular chat:  That link points to: http://web.archive.org/web/20080409050311/http://personalpages.manchester.ac.uk/staff/daryn.lehoux/Tropes.pdf  In the transcript, it looks like this:  That link points to: http://web.archive.org/web/20080409050311/%3Ca%20href=  ... which doesn't work so well! Could this be fixed? I'm not sure if this impacts links to elsewhere. I heard myself saying something was "tickety-boo", meaning good, successful, or satisfactory. Does anyone know where this strange-sounding phrase originated? Do you really want to create a new tag? How can top principal components retain the predictive power on a dependent variable (or even lead to better predictions)? Why did Littlefinger give Sansa to the Boltons? This question is about the TV show only.  In the books, Sansa is still with Littlefinger and everything that happens between Sansa and Ramsay in the show actually takes place between Ramsay and Jeyne Poole in the books.  However I don't recall why Littlefinger sends Sansa north. Apparently, in the books, it's safer to keep her traveling. So why did he do it in the show? The Boolean modifier is not working Can you connect to a LAN server without being on the same internet connection How can I use the MS JDBC driver with MS SQL Server 2008 Express? My configuration:     windows XP SP3 JDBC 2005 MS SQL Server 2008 Express, exposed via tcp/ip on port 1433 sqljdbc.jar in class path    I tried:  try {     Class.forName("com.microsoft.sqlserver.jdbc.SQLServerDriver").newInstance();     con = DriverManager.getConnection("jdbc:microsoft:sqlserver://localhost:1433/SQLEXPRESS2008;databaseName=Test;selectMethod=cursor", "sa", ""); } catch (Exception e) {     e.printStackTrace(); }   But it always throws an exception:  java.sql.SQLException: No suitable driver   I also tried the following urls:  localhost:1433/SQLEXPRESS2008  localhost/SQLEXPRESS2008  localhost   Same results.  Any help? "viewed 1 times" When a post is first created, it says "viewed 1 times" in the sidebar. This should say "viewed 1 time". Assume   $$\lim_{n\to \infty}{x_ny_n}=0$$  then  $$\lim_{n\to \infty}{x_n}=0$$ or $$\lim_{n\to \infty}{y_n}=0$$  I couldnt find two series who disprove that.. just hint please thanks Ramanujan log-trigonometric integrals I discovered the following conjectured identity numerically while studying a family of related integrals.  Let's set $$ R^{+}:=  \frac{2}{\pi}\int_{0}^{\pi/2}\sqrt[\normalsize{8}]{x^2 + \ln^2\!\cos x} \sqrt{ \frac{1}{2}+\frac{1}{2}\sqrt{ \frac{1}{2}+ \frac{1}{2}  \sqrt{ \frac{\ln^{2}\!\cos x}{ x^2 + \ln^2\! \cos x}}}}\,\mathrm{d}x, \tag1 $$ $$ R^{-}:=  \frac{2}{\pi}\int_{0}^{\pi/2}\frac{1}{\sqrt[\normalsize{8}]{x^2 + \ln^2\!\cos x}} \sqrt{ \frac{1}{2}+\frac{1}{2}\sqrt{ \frac{1}{2}+ \frac{1}{2} \sqrt{ \frac{\ln^{2}\!\cos x}{ x^2 + \ln^2\! \cos x}}}}\,\mathrm{d}x. \tag2 $$ We may numerically observe with at least 500 digits of precision that $$  \begin{align} &amp; R^{+}R^{-} \stackrel{?}{=}1 \tag 3 \\\\ &amp; R^{+} \stackrel{?}{=} \sqrt[\normalsize{4}]{\ln 2} \tag4 \\\\ &amp; R^{-} \stackrel{?}{=}\frac{1}{\sqrt[\normalsize{4}]{\ln 2}}. \tag5 \end{align} $$  How can we prove it?  A version of this has been sent to Eric Weisstein, these integrals are on Mathworld as . I have a group of objects of the same colour, consisting of compound paths, regular paths and groups of these. How to change the colour of all objects at once? When I tried to use colour picker or change it through color palette the colour remained the same but the graphics was rendered with a very tiny outline of the new colour, which I guess is not printable. Usually I had to change the colour of all objects individually.  Edit: I'm sorry if I was not too precise. I mean to apply new colour / swatch to all member objects of the group at once, not to change properties of a commonly shared swatch.    Edit 2: I received very useful answers, however my goal was to pick a colour from another object, serving as a source. When I tried to select all objects, for example with ctrl-a, and then use a eyedropper to pick up a desired colour, it didn't work for some objects in some groups. Please have a look at my answer where several methods are listed. I am trying to buy a laptop, and I have seen in many place that they show GDDR3 and sometimes  DDR3. What is the difference between them?  afaik DDR3 is a type of RAM used for system memory, while GDDR3 is a type of RAM used for graphics cards. So does "G" only stand for Graphics or does it have another particular meaning? How to sanity check a date in Java I find it curious that the most obvious way to create Date objects in Java has been deprecated and appears to have been "substituted" with a not so obvious to use lenient calendar.  How do you check that a date, given as a combination of day, month, and year, is a valid date?  For instance, 2008-02-31 (as in yyyy-mm-dd) would be an invalid date. Weird large K symbol C# "internal" access modifier when doing unit testing I'm new to unit testing and I'm trying to figure out if I should start using more of internal access modifier. I know that if we use internal and set the assembly variable InternalsVisibleTo, we can test functions that we don't want to declare public from the testing project. This makes me think that I should just always use internal because at least each project (should?) have its own testing project. Can you guys tell me why I shouldn't do this? When should I use private? Proof of $\gcd(a,b)=ax+by\ $ [Bezout's identity] Is there anything special with complex fraction $\left|\frac{z-a}{1-\bar{a}{z}}\right|$? Is there anything special with the form:  $$\left|\frac{z-a}{1-\bar{a}{z}}\right|$$ ? With $a$ and $z$ are complex numbers.  In fact, I saw it in a problem:   If $|z| = 1$, prove that $|\frac{z-a}{1-\bar{a}{z}}| = 1$ If $|z| &lt; 1$ and $|a| &lt; 1$, prove that $|\frac{z-a}{1-\bar{a}{z}}| &lt; 1$   I can easily prove the first one with expansion:  $$z=\cos\theta + i\sin \theta \\ a = m +in$$  But it will be terrible to use in the second one. What's more, I found this form a little special so maybe there is some clever trick without using expansion?  Thank you a lot! Is $\operatorname{lcm}(a,\gcd(b,c))=\gcd(\operatorname{lcm}(a,b),\operatorname{lcm}(a,c))$ AttributeError: MapDocObject: Unable to save on mxd.save/saveACopy? I'm trying to use arcpy to save an .mxd and am having an issue.  It doesn't matter if I use mxd.save or mxd.saveACopy I get the same error.  I know it doesn't look like I am doing anything w/ it here, but it's just a portion of my code:  import arcpy import arcpy.mapping as map arcpy.env.overwriteOutput = True  env.workspace = r"C:\MGIS\FinalProject2.gdb     mxd = map.MapDocument(r"C:\MGIS\Finalproject2.mxd") df = map.ListDataFrames(mxd)[0]  for df in map.ListDataFrames(mxd): df.scale = 2000000  #Add layers into map document addLayer = map.Layer(OKXcounties) addLayer1 = map.Layer(contourLabelText) addLayer2 = map.Layer(contoursClipLyr) addLayer3 = map.Layer(clipOKXLyr) map.AddLayer(df, addLayer, "AUTO_ARRANGE") map.AddLayer(df, addLayer1, "AUTO_ARRANGE") map.AddLayer(df, addLayer2, "AUTO_ARRANGE") map.AddLayer(df, addLayer3, "AUTO_ARRANGE")  # Add contour Labels for lyr in map.ListLayers(mxd):     if lyr.name == "contourLabelText":         lyr.showLabels = True     if lyr.name == "Contour Features":         lyr.transparency = 100  mxd.saveACopy(mxd)   I get the following error:   Traceback (most recent call last):   File "C:\Python26\ArcGIS10.0\Lib\site-packages\pythonwin\pywin\framework\scriptutils.py", line 312, in RunScript     exec codeObject in __main__.__dict__   File "C:\MGIS\geog485\FinalProject2\mxdtest.py", line 7, in &lt;module&gt;     mxd.saveACopy(mxd)   File "C:\Program Files\ArcGIS\Desktop10.0\arcpy\arcpy\utils.py", line 181, in fn_     return fn(*args, **kw)   File "C:\Program Files\ArcGIS\Desktop10.0\arcpy\arcpy\_mapping.py", line 668, in saveACopy     self._arc_object.saveACopy(file_name) AttributeError: MapDocObject: Unable to save.  Check to make sure you have write access to the specified file and that there is enough space on the storage device to hold your document.   Space is not the issue, I have 60 GB of free space and I've checked the windows properties of the .mxd file.  Read-only box is unchecked and I've "allowed" all permissions for all groups:    Does anyone have any idea how if this is a bug or how I can fix this?  I'm using ArcInfo 10.0 SP4.