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% $BNL;RJ*M}3X(B : $B1i=,LdBj(B 2-8 $B2rEz(B
% $B:n(B  $B@.(B  $B<T(B : 941000B $B8eF#(B  $BGn;V(B
% $B:n@.G/7nF|(B : /99/11/8 11/15
% $BEE;R%a!<%k(B : g941000@mail.ecc.u-tokyo.ac.jp
%
% \mondai{$B:n@.<T(B}{$B:n@.G/7nF|(B}{$BEE;R%a!<%k(B}{$BLdBjHV9f(B}{$BLdBjJ8(B}{$B2rEz(B}
%
\mondai
{941000B $B8eF#(B  $BGn;V(B}
{99/11/15}
{g941000@mail.ecc.u-tokyo.ac.jp}
{2-8}
{$B?eAG86;R$N%i%$%^%s7ONs(B($B=*>uBV!"(B$m$ = 1)$B$N%9%Z%/%H%k$NI}(B($B8w$N4QB,(B
$B$5$l$k%(%M%k%.!<$NI}(B)$B$H%P%k%^!<7ONs(B($m$ = 2)$B$N%9%Z%/%H%k$NI}$NHf$r(B
$B5a$a$h!#(B($BCm(B:$BHf$O!"%(%M%k%.!<$NI}$NHf$G5a$a$h!#(B)}
{
$BEE;R$,!"(B$n$$BHVL\$N%(%M%k%.!<=`0L$+$i(B$m$$BHVL\$N%(%M%k%.!<=`0L$KA+0\$9$k(B
$B$H$-$K!"?eAG86;R$O(B
\begin{equation}
E_{nm} = R(\frac{1}{m^2}-\frac{1}{n^2})
\end{equation}
$B$N%(%M%k%.!<$r;}$C$?8w$rJ|=P$9$k!#$3$3$G!"(B$R$$B$O%j%e!<%I%Y%jDj?t$G$"$k!#(B\\
$B$^$:!"%i%$%^%s7ONs$K$D$$$F!"%i%$%^%s7ONs$O!"=*>uBV$,(B$m$ = 1$B$G$"$k$+(B
$B$i!";O>uBV$O(B$n = 2,\quad 3,\quad 4 $B!E!E!E(B$$B$HJQ2=$9$k$3$H$,$G$-$k!#(B\\
$B$h$C$F!"8w$N%(%M%k%.!<$O!"(B
\begin{eqnarray}
E_{21} = R(\frac{1}{1^2}-\frac{1}{2^2}) = R(1-\frac{1}{4}) = \frac{3}{4}R{\ }\\
E_{31} = R(\frac{1}{1^2}-\frac{1}{3^2}) = R(1-\frac{1}{9}) = \frac{8}{9}R  
\end{eqnarray}
$B$NCM$r$H$k$3$H$+$i!"%(%M%k%.!<$N$H$j$&$kHO0O$O!"(B
\begin{equation}
\frac{3}{4}R \quad (n=2$B$N$H$-(B){\leq}  E < R \quad(n{\rightarrow} {\infty}$B$N$H$-(B)
\end{equation}
$B$G$"$k$3$H$,J,$+$k!#$h$C$F!"%i%$%^%s7ONs$N%9%Z%/%H%k$NI}$O!"(B
\begin{equation}
R - \frac{3}{4}R = \frac{1}{4}R 
\end{equation}
$B$G$"$k!#(B
$B$^$?!"%P%k%^!<7ONs(B($B=*>uBV!"(B$m$ = 2 $B;O>uBV!"(B$n = 3,\quad 4,\quad 5 $B!E!E!E(B )$$B$K$D$$(B
$B$F$bF1MM$K!"(B
\begin{eqnarray}
E_{32} = R(\frac{1}{2^2}-\frac{1}{3^2}) = R(\frac{1}{4}-\frac{1}{9}) = \frac{5}{36}R{\ } \\
E_{42} = R(\frac{1}{2^2}-\frac{1}{4^2}) = R(\frac{1}{4}-\frac{1}{16}) = \frac{3}{16}R
\end{eqnarray}
$B$H$J$k$+$i!"%(%M%k%.!<$N$H$j$&$kHO0O$O!"(B
\begin{equation}
\frac{5}{36}R \quad(n=2$B$N$H$-(B) {\leq}  E < \frac{1}{4}R \quad (n{\rightarrow}{\infty}$B$N$H$-(B)
\end{equation}
$B$G$"$j!"%P%k%^!<7ONs$N%9%Z%/%H%k$NI}$O!"(B
\begin{equation}
\frac{1}{4}R - \frac{5}{36}R = \frac{1}{9}R
\end{equation}
$B$H$J$k!#(B
$B$h$C$F!"%i%$%^%s7ONs$N%9%Z%/%H%k$NI}$H%P%k%^!<7ONs$N%9%Z%/%H%k$N(B
$BI}$NHf$H$7$F!"(B
\begin{equation}
\frac{1}{4}R : \frac{1}{9}R = 9 : 4
\end{equation}
$B$rF@$k!#(B}
