1-1/2+1/3-1/4+......+1/(2n-1)-1/2n<(2^1/2)/2
LHS=∑(1,2n)1/i - ∑(1,n)1/i=<∫(0,1)1/(1+x)=ln2<RHS

1-1/2+1/3-1/4+......+1/(2n-1)-1/2n=1/(n+1)+1/(n+2)+..+1/(2n)
由于(1+1/n)^n<e<(1+1/n)^(n+1)
所以1<ln(1+1/n)*(n+1) 即ln(1+1/n)>1/(n+1)
有LHS<ln(1+1/n)+ln(1+1/(n+1))+..+ln(1+1/(2n-1))=ln(2n/n)=ln2<sqrt(2)/2