🚀 go-pugleaf

$BJ?2l!w?^=q4[>pJsBg3X$G$9!#(J
(1/2)^(1/3)^(1/4)^...^(1/n)^... $BLdBj$K$D$$$F$N$3$3$^$G$N$^$H$a$G$9!#(J

# Subject: $B$b(J tower series, Re: [Q];a problem of limit $B$N#2$D$r(J
# $B@8$8$5$;$F$7$^$C$F$9$_$^$;$s!#(J

$B0J2<$O$$$N$&$($5$s(J(inoue@ma.ns.musashi-tech.ac.jp (INOUE Koichi))
$B5Z$SJ?2l!JB>$K<c>>$5$s$J$I!K$N$3$3$^$G$N8!F$$r$^$H$a$?$b$N$G$9!#(J
$B5-9f$OJ?2l$NMQ$$$?$b$N$r;H$o$;$F$b$i$$$^$9!#(J

* $BH/C<!'(J(1/2)^(1/3)^...^(1/n)^... $B$N<}B+@-(J

* f(n) $B$r<+A3?t>e$N!J@5CM!K<B?tCM4X?t$H$9$k!#(J
$B!!$D$^$j(J f(1), f(2), ..., f(n), ... $B$OIaDL$G$$$&?tNs$G$"$k!#(J
  T(f;n) = f(1)^f(2)^...^f(n) $B$H$9$k!#(J
  $B$^$?7A<0E*$K!"(JT(f) = T(f;$B!g(J) = f(1)^f(2)^...^f(n)^... $B$H=q$/!#(J

* $B:#LdBj$K$7$?$$$N$O!"FC$K$9$Y$F$N(J n $B$K$D$$$F(J 0<f(n)<1 $B$N>l9g$G$"$k!#(J
  $B$3$N$H$-<!$,@.$jN)$D!'(J
  0 <T(f;1)=f(1)<T(f;3)<...<T(f;2k+1)<... <T(f;2k)<...<T(f;4)<T(f;2)< 1
  $B6v?tNs!"4q?tNs$O$H$b$KM-3&C1D4$@$+$i!"(J
	T(f;1)<T(f;3)<...T(f;2k+1) ->$B&A(J(f)
	T(f;2)>T(f;4)>...T(f;2k)   ->$B&B(J(f)
  $B$H$$$&6K8B$r;}$D!#(J
  $B$3$3$G(J $B&A(J(f)=$B&B(J(f) $B$J$i(J T(f) $B$O<}B+!"&A(J(f)<$B&B(J(f) $B$J$i(J T(f) $B$OH/;6$G$"$k!#(J

  $B$^$?(J I(f)=($B&A(J(f),$B&B(J(f)) $B$H=q$/!J(JI(f) $B$O3+6h4V$G!"&A(J(f)=$B&B(J(f) $B$J$i6u=89g!K!#(J
  $B>e$+$i!"(JT(f;n) \in I(f) $B$H$J$k(J n $B$OB8:_$7$J$$!#(J

* $BFC$K(J f(n) $B$,Dj?t4X?t$N>l9g!'$D$^$j(J a=f(1)=f(2)=...=f(n)=... $B$N>l9g!#(J
  T(a) $B$O!'(J
	A.e^(1/e)<a $B$J$i!g$KH/;6(J
	B.0.065... = (1/e)^e$B!e(Ja$B!e(Je^(1/e) = 1.444... $B$J$i<}B+(J
	$B!!6K8BCM$O!"(Jx=a^x $B$rK~$?$9(J x $B$H$7$FM?$($i$l$k!#(J
	  $BFC$K(J a=1 $B$N$H$-(J T(1;n) $B$O>o$K(J 1 $B$KEy$7$$!#(J
	C.0<a<(1/e)^e $B$J$i?6F0$7$J$,$iH/;6(J
	$B!!$3$N$H$-(J T(f) $B$O(J $B&A(J(a), $B&B(J(a) $B$H$$$&2<6K8B!">e6K8B$r;}$A!"(J
	$B!!(JT(f;n) $B$O$=$l$K8~$1$F?6F0$9$k!#(J
	$B!!$5$i$K(J a=0 $B$b2C$($k$H!"(JT(0) $B$O(J 0,1 $B$N4V$r?6F0$9$k!#(J

$B!!$J$*(J (1/e)^e $B$O0J2<(J e^(-e) $B$H=q$/!J#1;z8:$k!*!K(J
$B!!(JC. $B$N$H$-!"(Jc(a)=a^c(a) $B$H$J$k(J c(a) $B$bB8:_$7!"(J
$B!!&A!"&B!"(Jc $B$O<!$N$h$&$J4X78$rK~$?$9!#(J
	$B&A(J(a)<c(a)<$B&B(J(a)
	a^$B&A(J(a) = $B&B(J(a)
	a^$B&B(J(a) = $B&A(J(a)
	$B$3$l$+$i(J $B&A(J^$B&A(J = $B&B(J^$B&B!"&A(Jlog$B&A(J = $B&B(Jlog$B&B$J$I$b!#(J
$B!!$^$?(J $B&A(J(a) $B$O(J a $B$KBP$7$FC1D4A}2C!"&B(J(a) $B$O(J a $B$KBP$7$FC1D48:>/!#(J
	# $B$3$N>ZL@$O<c43$NOSNO$,I,MW(J
$B!!!J$J$*(J a->e^(-e) $B$N$H$-!"(Jlim $B&A(J(a) = lim $B&B(J(a) = lim c(a) = 1/e$B!K(J

* $B>e$+$i!"(J0<f(n)<e^(-e) $B$G!"(Jf(n) $B$,!J9-5A!KC1D48:>/$J$i!"(JT(f) $B$OH/;6$9$k!#(J

* $BG$0U$N(J N $B$K$D$$$F!"(JT(f) $B$N<}B+@-$O!"$=$N(J N+1 $B9`0J9_$N(J
	T(f;N,$B!g(J) = f(N+1)^f(N+2)^...
$B!!$N<}B+@-$HF1$8!J(JT(f;N,$B!g(J) $B$,<}B+!?H/;6(J <===> T(f) $B$,<}B+!?H/;6!K(J
$B!!$?$@$7(J f(i) $B!b(J 0,1 (1$B!e(Ji$B!e(JN) $B$H$9$k!#(J

* $B>e#2E@$+$i!"(J1) 0<f(n)<1$B!"(J2) f(n) $B$OC1D48:>/!"(J3) n>N $B$J$i(J f(n)<e^(-e)
$B!!$H$J$k(J N $B$,B8:_$9$k$J$i!"(JT(f) $B$OH/;6$9$k!#(J
$B!!FC$KEv=iLdBj!'(J(1/2)^(1/3)^... = T(1/(n+1)) $B$OH/;6$9$k!#(J
$B!!F1MM$K(J 1>f(1)$B!f(Jf(2)$B!f(J...$B!f(Jf(n)$B!f(J... -> 0 $B$G$"$k!'(J
	(1/2)^(1/4)^(1/8)^...
	(1/sqrt(2))^(1/sqrt(3))^(1/sqrt(4))^...
$B!!$J$I$bH/;6$9$k!#(J

* 0<f(n)<a<e^(-e) $B$J$i(J I(a) << I(f) $B!J(JI(a) $B$O(J I(f) $B$NItJ,=89g$N0U!K$H$$$&(J
$B!!Ev=i$NJ?2lM=A[$OH]Dj$5$l$?!#(J
$B!!$3$l$,8@$($l$P(J T(f) $B$OH/;6$J$N$@$,!"$7$?$,$C$F$3$NJ}8~$G$O%@%a!#(J
$B!!(JT(f) $B$OH/;6$+$H$$$&LdBj<+BN$OL$2r7h!J$?$V$s@5$7$$!"$H$$$&$N$,J?2lM=A[!K!#(J

* $B$=$NB>0lHL$N%1!<%9$J$I$K$D$$$F$O$[$H$s$I2?$b$o$+$C$F$$$J$$!#(J

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