I'm trying to demonstrate the convergence of the following series: $$\sum \frac{\sin(1/n)}{\sqrt{n}}$$  I know I can prove the convergence by the comparison test. I started like  $$\frac{\sin(1/n)}{\sqrt{n}} \le \frac{1}{\sqrt{n}},$$  but now I'm stuck because $\frac{1}{\sqrt{n}}$ diverges. With which sequence can I compare it to be able to use the comparison test?  Can you give me a hint?
So my teacher gave a pretty tough (for me) problem in class today: Does the series $$ \sum_{n=1}^{\infty}\frac {\sin{\frac 1 n}} {\sqrt n} $$ converge or diverge?  So far I've thought of trying the Comparison Test and Ratio Test but couldn't really get anywhere. Intuitively I think it should converge, but how do I actually show this?
Story about building up speed leading to increased quality of life Can linear momentum be conserved before and after collision in the presence of an external force? I'm a physics tutor for 11th and 12th grade students. I taught my students that momentum is conserved only when the net external force on the system is zero. But in the books we are following, momentum was conserved in the following cases in spite of presence of an external force:   A body (mass m1) is dropped from a tower (height h) and another body (mass m2) is projected vertically upward with initial velocity u. Find the velocities just after collision. (Gravitational force is the external force) Two bodies projected towards each other on a rough table surface, with all required parameters given asked to find the velocities after collision. (frictional force is the external force) Body explodes in mid air. Momentum is conserved before and after explosion. (Again gravitational force is the external force)   I explained to my students in all these cases though an external force is present it has negligible effect. But I'm not satisfied with my answer.  EDIT: Thank you for the comments and answer. It is clear that contribution of external force to change in momentum is negligible. Can someone do the calculation of taking change in momentum into consideration and showing that it is negligible? How come the year doesn't show here? The selected task "{0}" no longer exists. To see the current tasks, click Refresh When I start my task scheduler on Windows 7 I get a dialog with the following error:     The selected task "{0}" no longer exists.  To see the current tasks,   click Refresh     Also there's a UUID, that is displayed as the name and when I try to delete it I cannot, it just hangs around.  Is there a fix for this? error tex capacity exceeded log files added I am having a hard time figuring out where the problem is with my code/workflow. I previously posted this question , but then the same problem came straight back. When I run it in tex, I get an error input limit exceeded. The exact log is pasted below:  This is pdfTeX, Version 3.14159265-2.6-1.40.17 (MiKTeX 2.9.6210 64-bit) (preloaded format=pdflatex 2017.1.26)  26 JAN 2017 20:45 entering extended mode **./master.tex (master.tex LaTeX2e &lt;2016/03/31&gt; patch level 3 Babel &lt;3.9r&gt; and hyphenation patterns for 75 language(s) loaded. ("C:\Program Files\MiKTeX 2.9\tex\latex\base\article.cls" Document Class: article 2014/09/29 v1.4h Standard LaTeX document class ("C:\Program Files\MiKTeX 2.9\tex\latex\base\size11.clo" File: size11.clo 2014/09/29 v1.4h Standard LaTeX file (size option) ) \c@part=\count79 \c@section=\count80 \c@subsection=\count81 \c@subsubsection=\count82 \c@paragraph=\count83 \c@subparagraph=\count84 \c@figure=\count85 \c@table=\count86 \abovecaptionskip=\skip41 \belowcaptionskip=\skip42 \bibindent=\dimen102 ) ("C:\Program Files\MiKTeX 2.9\tex\latex\apacite\apacite.sty" Package: apacite 2013/07/21 v6.03 APA citation \c@BibCnt=\count87 \bibleftmargin=\skip43 \bibindent=\skip44 \bibparsep=\skip45 \bibitemsep=\skip46 \biblabelsep=\skip47 ) No file master.aux. \openout1 = `master.aux'.  LaTeX Font Info:    Checking defaults for OML/cmm/m/it on input line 5. LaTeX Font Info:    ... okay on input line 5. LaTeX Font Info:    Checking defaults for T1/cmr/m/n on input line 5. LaTeX Font Info:    ... okay on input line 5. LaTeX Font Info:    Checking defaults for OT1/cmr/m/n on input line 5. LaTeX Font Info:    ... okay on input line 5. LaTeX Font Info:    Checking defaults for OMS/cmsy/m/n on input line 5. LaTeX Font Info:    ... okay on input line 5. LaTeX Font Info:    Checking defaults for OMX/cmex/m/n on input line 5. LaTeX Font Info:    ... okay on input line 5. LaTeX Font Info:    Checking defaults for U/cmr/m/n on input line 5. LaTeX Font Info:    ... okay on input line 5. \c@maskedRefs=\count88 ("C:\Program Files\MiKTeX 2.9\tex\latex\apacite\english.apc" File: english.apc 2013/07/21 v6.03 apacite language file LaTeX Info: Redefining \BPBI on input line 129. LaTeX Info: Redefining \BHBI on input line 130. ) ! TeX capacity exceeded, sorry [input stack size=5000]. \AtBeginDocument #1-&gt;#1  l.5 \begin{document}  If you really absolutely need more capacity, you can ask a wizard to enlarge me.   Here is how much of TeX's memory you used:  641 strings out of 493333  7639 string characters out of 3139188  74825 words of memory out of 3000000  4254 multiletter control sequences out of 15000+200000  3940 words of font info for 15 fonts, out of 3000000 for 9000  1141 hyphenation exceptions out of 8191  5000i,0n,4998p,136b,36s stack positions out of 5000i,500n,10000p,200000b,50000s !  ==&gt; Fatal error occurred, no output PDF file produced!   I previously posted this question and thought I solved it but am having the same problem again.   The latex code I am trying to execute is below:   \documentclass[11pt,twoside]{article} \usepackage{apacite} \nocite{*}  \begin{document}   % title page. Edit as needed for future publications.  \title{Research Proposal} \author{Teererai Marange \\ Department of Computer Science \\ University of Auckland\\ \texttt{t.marange@auckland.ac.nz}} \date{January 26, 2017} \maketitle % end title page.    % abstract goes here \newpage \begin{abstract} Abstract goes here  \end{abstract} \newpage   % table of contents  \tableofcontents \newpage   %begin introduction  \section {Introduction} Lorem Ipsum Doremifaso   %\cite{Nedakeyword} use to cite     %references our biliography. This must go whereever we use our bibtex references  \bibliography{references} \bibliographystyle{apacite}   \end{document}   and the bib file that I am using to build the bibliography is below:   @misc{wikipedia1,   title = "Scripting language",   author = "Wikipedia",   url = {http://en.wikipedia.org/wiki/Scripting_language},   year = 2014, } How are reserved and formal synonyms? In the dictionary reserved means - not revealing emotions and formal means - prim or stiff. I saw they were synonym in the Google dictionary. What makes these two words similar, I wonder? What event log should I check for Hardware Changes I would like to obtain a realistic rendering with edges not too sharp and smooth surface. I tried adding a crease of 1, but edges are too sharp. Marking edges sharp is too soft. It seems there are some artifacts in the rounded corners...what can I do? What are some ways to write object-oriented code in C? Especially with regard to polymorphism.    See also this Stack Overflow question . How to remove "phone style" Gnome swipe to unlock Why couldn't Harry see Thestrals at the end of Goblet of Fire? Why couldn't Harry see Thestrals at the end of Goblet of Fire after the senseless and shocking death of Cedric Diggory?   At the end of Goblet of Fire, the carriages still appeared horseless to Harry:     Hermione turned away, smiling at the horseless carriages which were now trundling towards them up the drive, as Krum, looking surprised, but gratified, signed a fragment of   parchment for Ron.   Goblet of Fire - Page 629 - British Hardcover - Adult Edition   At the beginning of Order of the Phoenix, Harry sees the Thestrals for the first time:     Here stood the hundred or so horseless stagecoaches that always took the students above first year up to the castle. Harry glanced quickly at them, turned away to keep a lookout for Ron and Hermione, then did a double-take.   The coaches were no longer horseless. There were creatures standing between the carriage shafts.   Order of the Phoenix - Page 178 - British Hardcover - Adult Edition   We know that only those who have seen death can see Thestrals:     ‘Don’ worry, it won’ hurt yeh,’ said Hagrid patiently. ‘Righ’, now, who can tell me why some o’ yeh can see ’em an’ some can’t?’   Hermione raised her hand.   ‘Go on then,’ said Hagrid, beaming at her.   ‘The only people who can see Thestrals,’ she said, ‘are people who have seen death.’   ‘Tha’s exactly right,’ said Hagrid solemnly [...]   Order of the Phoenix - Page 394 - British Hardcover - Adult Edition   A month went by between Cedric's death and the end of the spring term -- is this enough time for Harry to have truly internalized Cedric's death? Goblet of Fire indicates that Harry was in shock following Cedric's murder, but was able to relate the story of Cedric's death to Mr. and Mrs. Diggory.     When he looked back, even a month later, Harry found he had few memories of the following days. It was as though he had been through too much to take in any more. The recollections he did have were very painful. The worst, perhaps, was the meeting with the Diggorys that took place the following morning.   [...]   ‘He suffered very little, then,’ she said, when Harry had told her how Cedric had died.'   Goblet of Fire - Page 621 - British Hardcover - Adult Edition   So, then, why couldn't Harry see Thestrals at the end of Goblet of Fire? Is there a program equivalent to lint for LaTeX? Is there a program equivalent to lint for LaTeX? (lint checks C code for syntax errors and possible mistakes.) Labelled in-line equation What is JavaScript's highest integer value that a number can go to without losing precision? Is this defined by the language? Is there a defined maximum? Is it different in different browsers? What is ground and what does it do? What is the use of the Dot Product of two vectors? Push Notification Badge Count Not Updating After taking a statistics course and then trying to help fellow students, I noticed one subject that inspires much head-desk banging is interpreting the results of statistical hypothesis tests.  It seems that students easily learn how to perform the calculations required by a given test but get hung up on interpreting the results.  Many computerized tools report test results in terms of "p values" or "t values".  How would you explain the following points to college students taking their first course in statistics:   What does a "p-value" mean in relation to the hypothesis being tested?  Are there cases when one should be looking for a high p-value or a low p-value? What is the relationship between a p-value and a t-value? Invalid selectOptions found. Use SelectOption type in Apex Now that I have Google Chrome all set up, I got myself a new laptop. Does anyone have any idea how to transfer my settings, cookies, favorites, etc?   I checked documents and settings\me\application data, and found nothing.