I was asked this question in a PhD interview. They didn't say whether $a_n$ is non-increasing or strictly decreasing. I tried various examples such as $a_n=\frac 1{n^p}$ where $p \gt 1$, $a_n=\frac 1{n^2 \ln n}$. Then $n a_n \to 0$.  I couldn't answer this question then.  But when I saw Robert Israel's answer to  question, motivated from it, I defined a sequence $a_n$ as below :  $ a_n = \begin{cases} \frac 1n,  &amp; \text{if $n$ is a power of 2} \\ 0, &amp; \text{otherwise.} \end{cases}$  Here, $\sum a_n=\frac 12+ \frac 14+ \frac 18+ \cdots =\frac {\frac 12}{1 -\frac 12}=1$ and $\{n a_n\}$ diverges, because there are infinitely many terms of $0$ and $1$ in it. i.e.  $n a_n = \begin{cases} 1,  &amp; \text{if $n$ is a power of 2} \\ 0, &amp; \text{otherwise.} \end{cases}$  So $\{na_n\}$ converges in some cases and diverges in some cases. Am I right?  Note : I have gone through these posts, ,,  too but they have some extra conditions or specific details.  Thanks.
Can it be proven that : If $a_n&gt;0$ and $\sum a_n$ converges then $na_n \to 0$ (without assuming $a_n$ is decreasing)  Please note This question does not require $a_n$ to be decreasing as condition,  Is it possible to prove $na_n \to 0$ without requiring $a_n$ to be decreasing? I keep thinking that $a_n&gt;0$ and $\sum a_n$ converging then that implies that $a_n$ must be decreasing anyway(not monotonically decreasing necessarily )  Is it possible that $a_n&gt;0$ and $\sum a_n$ converges then $na_n \not \to 0$?  There are already answered questions that assume $a_n$ to be decreasing as a requirement: (this question does not assume $a_n$ is decreasing)       was the post that made me ask this question, a counter example of monotonoic decreasingness of $a_n$ was given in comments by the OP.
How can I change the default date format? When I paste a date into Excel 2010 in the format "2012-12-07 00:00:00.000" I'd like it to be displayed by default as "2012-12-07", but it is actually displayed as "00:00.0".   I know I can work around this by prepending a ' or changing the cell format, but I'm interested in changing the default format.  An approach that almost works is going to Control Panel->Region and Language->Long Time format and changing it to something like "h". This results in Excel displaying the dates I paste in as "2012-12-07 00:00:00.000", but screws up times displayed outside of Excel.  The dates I'm pasting in are from SQL Server. Proving that the set of limit points of a set is closed Subgroup of $\mathbb{R}$ either dense or has a least positive element? How can I get a specific Floor in my Tower? I really really want a specific floor () in my Tiny Tower.  How does the game decide which exact business will go into the "Creative" floor?  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David   Webb    Box 34 Rural Route 2    Nixa    MO  65714   (417)555-1478   555-66-7788 09-13-1970 Martha  Kent    1122 North Hwy 5    Smallville  KS  66789   (785)555-2322   343-55-8845 04-17-1965 Edward  Nygma   443 W. Broadway Gotham City NJ  12458   (212)743-3537   785-48-5524 08-08-1987 O'Reilly    Baba    123 Torch Ln    Joplin  MO  64801   (417)222-1234   456-65-3211 12-13-1999 Martin Bob  44 Boss Rd  Joplin  MO  64801   (417)888-4565   456-98-1111 01-01-2007   the dates are in the 9th field and I want to display them as January 7, 2017 instead of 01-07-2017 for example.  How should I do that? If using options please explain what they do briefly. Doing this in bash. Needing to put it in a script and output to a new file to preserve original. Can you intentionally fail a skill check? 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SQL injection that gets around mysql_real_escape_string() "The ones" or "those"? I recently corrected  &quot;ghettos, such as the ones found...&quot; to &quot;ghettos, such as those found...&quot;  Was I correct, or are both versions right?