I know one solution.   That is by Doing comparison with $\dfrac{4^n+4^n}{6^n}$  Wondering if there are more ways to do it
I am trying to determine if the sum $\sum_{n=1}^\infty \frac{n+4^n}{n+6^n}$ is convergent or divergent.   In attempting to solve this problem I began by seeing if I could do the divergence test - in this case it is not helpful. The terms $4^n,6^n$ look like geometric series type terms but the $n$ in the numerator and denominator are troublesome. I first tried factoring $n$ out of the expression:  $$\sum_{n=1}^\infty \frac{n+4^n}{n+6^n} = \sum_{n=1}^\infty \frac{1+\frac{4^n}{n}}{1+\frac{6^n}{n}}$$  Next I tried comparing to the series $$\sum_{n=1}^\infty \frac{\frac{4^n}{n}}{\frac{6^n}{n}} = \sum_{n=1}^\infty \frac{{4^n}}{{6^n}} = \sum_{n=1}^\infty{\left(\frac{2}{3}\right)}^n$$  This is a convergent geometric series as $r = \frac{2}{3}\lt1$  now I reason that the series $\sum_{n=1}^\infty \frac{1+\frac{4^n}{n}}{1+\frac{6^n}{n}}$ is neither smaller not greater than the series $\sum_{n=1}^\infty{(\frac{2}{3})}^n$ Am I able to conclude anything in this case by a direct comparison test or can I only conclude that the first series is convergent if the second is greater than the first?  Thanks
Adding shapefile or feature class as layer in ArcMap using ArcPy I am trying to automate various tasks in ArcGIS Desktop (using ArcMap generally) with Python, and I keep needing a way to add a shapefile to the current map. (And then do stuff to it, but that's another story).  The best I can do so far is to add a layer file to the current map, using the following ("addLayer" is a layer file object):  def AddLayerFromLayerFile(addLayer):  import arcpy  mxd = arcpy.mapping.MapDocument("CURRENT")  df = arcpy.mapping.ListDataFrames(mxd, "Layers")[0]  arcpy.mapping.AddLayer(df, addLayer, "AUTO_ARRANGE")  arcpy.RefreshActiveView()  arcpy.RefreshTOC()  del mxd, df, addLayer   However, my raw data is always going be shapefiles, so I need to be able to open them. (Equivalently: convert a shapefile to a layer file without opening it, but I'd prefer not to do that). Add desktop notifications for Chrome or Firefox Why do we say "try and" [verb] instead of "try to" [verb]? (E.g., "Try and call me tomorrow.") In written and standard semi-formal (and above) spoken English, one would use &quot;try to&quot;:  Try to be a better person. Try to get the fishhook out of my thumb, please. Try to find a pharmacy when you need one.  But in spoken English, we (Americans, at least) usually substitute &quot;try and&quot; for all those cases. &quot;Try to&quot; makes sense because the &quot;to&quot; begins the construction of an infinitive: &quot;try to be&quot;, &quot;try to do&quot;, &quot;try to act&quot; and so on. &quot;Try and&quot; on the other hand doesn't seem to make any sense. I'm curious how the &quot;try and&quot; crept in, and when. It's really tough to Google small words like these, so I'm not finding anything on the Web. Is it a contraction of something like &quot;Try hard, and ...&quot;? Note: I've seen  and it is somewhat related but doesn't tell me what I want to know. Edit Observe how, even on this site,  construction. 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Does this mean he will be available on the morning of the 15th? Comparison theorem for ODE Set up BIND9 as DNS Firewall In XFCE4 desktop environments, alt+scroll wheel zooms my entire desktop. I'm finding this very annoying because sometimes I flick the scroll wheel to go to the top of a document then press the alt key for a some unrelated keyboard shortcut and find myself zoomed in so far I can't see anything at all. Is there any way to disable this feature? Allow people to answer protected questions if they have earned at least some amount of combined SE rep The purpose of protected questions is, according to the message presented on such a question, "to prevent 'thanks!', 'me too!', or spam answers by new users." The more verbose  agrees on the purpose. And the  says this is why protected questions exist:     Some questions are protected because they are expected to attract   either spam or users -- often new users -- who may mistake the site as   a traditional forum, posting "noisy" answers such as "Thank you" or   "This worked for me" or "I'm also having this problem".   Considering this, shouldn't a user be able to answer protected questions if they have gained at least 10 rep on any stack exchange site? If they've already demonstrated one site that they are familiar enough with the SE Q/A format to not be a drive-by useless answerer, shouldn't that confidence in them carry over to other SE sites?  Or maybe at some level of combined SE rep this restriction should be waived. I find it silly that at 15k+ combined network rep, I'm unable to answer a protected question on a SE site I've just started using. Roman numerals become “too wide” in table of contents Must $k$-subalgebra of $k[x]$ be finitely generated? What is the maximum length of a URL in different browsers? What is the maximum length of a URL in different browsers? Does it differ among browsers? Is a maximum URL length part of the HTTP specification?