if the limit $$L=\lim_{n\to \infty}\sqrt n \int_0^1 \frac{1}{(1+x^2)^n}dx$$exists and is larger than $\frac{1}{2}$ then  prove that $\frac{1}{2} &lt; L &lt; 2 $ .
$$  \lim_{n\to\infty} n^{1/2} \int_{0}^{1} \frac{1}{(1+x^2)^n}\mathrm{d}x=0 $$  Is my answer correct? But I am not sure of method by which I have done.
Expandably change letter case and use inside \csname, without a package I want to change the letter-case of a single character argument, and use that changed letter inside a \csname.  And I want to do it without an additional package.  For example, I want to pass a macro the letter {S}, and have it execute the macro \csname macros\endcsname, or pass it the letter {T}, and have it execute the macro \csname macrot\endcsname.    In my MWE, I show three approaches:   I didn't expect \lowercase to work, since I recall reading that it is not truly expanded until the last possible moment.  Surprisingly, it goes into an \edef okay, but it must not be truly expanded, because it breaks the \csname. I also formulated an approach where I used \numexpr to add 32 to the ASCII code of the argument and perform a \char upon that result.  Like \lowercase, it gives the appearance of being expandable, but it also fails inside the \csname. I finally achieve success with the stringstrings package.  Having written the package, I know the conversion was not pretty... essentially, it is an exhaustive 26-fold check of the form \if #1Aa\else\if #1Bb\else\if #1Cc\else...\fi\fi\fi, which has the virtue that it can be expandably placed into an \edef, and so it works.   There has to be a better way.  The MWE works only because I use the inefficiently calculated \tmp from METHOD 3 inside the \csname.  If I use \tmp from either of the two earlier methods, it breaks.  \documentclass{article} \usepackage{stringstrings} \def\macros{Expandably converted an S to a s} \def\macrot{Expandably converted a T to a t} \newcommand\convert[1]{%   %MAKE THE ARGUMENT LOWER CASE   1) LOWERCASE\\   \edef\tmp{\lowercase{#1}}% PRETENDS TO EXPAND, BUT FAILS IN \csname   Here is the ``expanded'' conversion: \tmp\\   % OR   2) ASCII CONVERSION\\   \edef\tmp{\expandafter\char\numexpr`#1+32\relax}% PRETENDS TO EXPAND, BUT FAILS IN \csname   Here is the ``expanded'' conversion: \tmp\\   % OR   3) STRINGSTRINGS PACKAGE\\   \caselower[q]{#1}\def\tmp{\thestring}% WORKS BUT USES A PACKAGE   Here is the ``expanded'' conversion: \tmp\\   %   Now use it in a csname: \csname macro\tmp\endcsname% } \begin{document} \convert{S}\par \convert{T} \end{document}       SUMMARY OF ANSWERS:  I have added egreg's code snippets into this MWE to produce a document (and not just terminal output), showing the methods put forth to this point:  \documentclass{article}  %%%%%%%%%%%%%%%%%%%%%%%%%%%% L3 APPROACH  \usepackage{expl3}  \ExplSyntaxOn \cs_set_eq:NN \Xexplower \tl_expandable_lowercase:n \ExplSyntaxOff  %%%%%%%%%%%%%%%%%%%%%%%%%%%% EGREG'S LESS HACKY SOLUTION  \def\explowerchar#1{%   \ifcase\numexpr`#1-`A\relax    a\or b\or c\or d\or e\or f\or g\or h\or i\or j\or k\or l\or m\or    n\or o\or p\or q\or r\or s\or t\or u\or v\or w\or x\or y\or z\else    #1\fi }  \def\explower#1{%   \doexplowerchar#1\relax }  \def\doexplowerchar#1{%   \ifx#1\relax   \else     \explowerchar{#1}\expandafter\doexplowerchar   \fi }  %%%%%%%%%%%%%%%%%%%%%%%%%%%% STEVE'S HACKY PACKAGE APPROACH  \usepackage{stringstrings}  %%%%%%%%%%%%%%%%%%%%%%%%%%%%  \def\macros{Expandably converted an S to a s} \def\macrot{Expandably converted a T to a t}  \begin{document}  \textbf{Steve's hacky approach}  \caselower[q]{S}\csname macro\thestring\endcsname\par \caselower[q]{T}\csname macro\thestring\endcsname  \textbf{egreg's less hacky approach}  \edef\lcstring{\explowerchar{S}}\csname macro\lcstring\endcsname\par \edef\lcstring{\explower{T}}\csname macro\lcstring\endcsname  \textbf{L3 approach}  \edef\lcstring{\Xexplower{S}}\csname macro\lcstring\endcsname\par \edef\lcstring{\Xexplower{T}}\csname macro\lcstring\endcsname  \end{document} Divisibility Proof with Induction - Stuck on Induction Step I'm working on a problem that's given me the run around for about a weekend. 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The pdf of $\frac{X_1-\bar{X}}{S}$ Suppose $X_1, X_2,...,X_n$ be i.i.d from $N(\mu,\sigma^2)$ with unknown $\mu \in \mathcal R$ and $\sigma^2&gt;0$  Let $Z=\frac{X_1-\bar{X}}{S},$ S is the standard deviation here.  It is can be shown that  $Z$ has the Lebesgue p.d.f.  $$f(z)=\frac{\sqrt{n} \Gamma\left(\frac{n-1}{2}\right)}{\sqrt{\pi}(n-1)\Gamma\left(\frac{n-2}{2}\right)}\left[1-\frac{nz^2}{(n-1)^2}\right]^{n/2-2}I_{(0,(n-1)/\sqrt{n})}(|Z|)$$  My question is then how to get this pdf?  The question is from  in example 3.3.4  to find the UMVUE of $P(X_1 \le c)$. I can understand the logic and procedures to find the UMVUE but don't know how to get the pdf.  I think this question also relate to this    Thank you very much for help or point to any related references will be also appropriated. So, I mount a 200mm lens on my Canon 450D. It effectively becomes a 320mm lens. Is this the equivalent of 320mm on a full frame camera? 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