\magnification=\magstep2
\hsize 6 true in
\centerline{\bf MY STORY}
\centerline{\sl by Albert Lai}
\vskip 0.5 in
Once upon a time, there was an android called Albert.
{\sl (An android is something like {\bf Lt.\ Cmdr.\ Data}, if you have watched
{\bf ST:TNG}.)}
He liked to apply logic and mathematics to {\sl every}thing in {\sl every}day
life.  MATB43, MATB31, MATB42, $\ldots$ all math he had learned.

\def\Emoticon{Em\"otic\'on}
Now, there was also this \Emoticon\ guy, who worked with emotions and
no logic.  So when \Emoticon\ met Albert, they crashed into each other.
Albert was mad at \Emoticon\ because \Emoticon\ didn't understand
Albert; and \Emoticon\ was mad at Albert because Albert didn't
understand \Emoticon.

Confusing, eh?

({\sl BTW, how could such a logician as Albert become mad?} :-)

\smallskip
One day Albert did the following math problem:
$$\int _0 ^1 f(x) \, dx \quad .$$
\Emoticon\ said, ``Oh, it's neat!  The answer is ONE!''  Albert said,
``How do you know the answer is $1$, while $f(x)$ is not even known?''
\Emoticon\ replied, ``Because {\sl I} feel like it!  You have a
problem?''

And then, the next day, \Emoticon\ did the following math problem:
$$\sum _{n=0}^{k} \int_0^1 {(-1)}^{k-n} {k \choose n} x^{k-n} \, dx \quad .$$
Albert said, ``Oh, it's easy!  The answer is $-1\over (k+1)$!''
\Emoticon\ said,
``How do you know it is the answer?  It is such a mess!  I think {\sl even}
the prof.\ himself can't do it!''  Albert replied, ``Elementary, \Emoticon!
Elementary!  It's logic!
$$\eqalign{
    & \sum _{n=0}^{k} \int_0^1 {(-1)}^{k-n} {k \choose n} x^{k-n} \, dx \cr
    = & \int_0^1 {(1-x)}^k dx \cr
    = & \left. {{(1-x)}^{k+1} \over k+1} \right| _0^1 \cr
    = & - {1 \over k+1} \quad ,\cr
} $$
which is to be demonstrated.  {\it Quad erat demonstratum\/}!''

\bye