证明:U=1/2+1/2^(2^2)+...+1/2^(n^2)+...为无理数.
反证法:设U=p/q,取2^(2n)>q,
==>
1/[q2^(n^2)]≤p/q-[1/2+1/2^(2^2)+...+1/2^(n^2)]=
=s/[q2^(n^2)]=1/2^[(n+1)^2)]+...+1/2^[(n+k)^2)]+...<
<1/2^[n^2+2n+1]+...+1/2^[n^2+2kn+1]+...
==>
1/q<1/2^[2n+1]+...+1/2^[2kn+1]+...=
=1/2^(2n+1){1+1/2^[2n]...+1/2^[2(k-1)n]+...}<
<1/2^(2n)矛盾.
所以U为无理数.
补:
0<p/q-[1/2+1/2^(2^2)+...+1/2^(n^2)]=p/q-t/2^(n^2)=
=s/[q2^(n^2)]==>s≥1.
