Question 2
a) Let $a_n=\ln(2n^2+1)-\ln(n^2+n+1)$ for $n=1,2,\ldots$. Prove that only finitely many $n$ satisfy $\{a_n\}\lt 1/2$.
b) Let $b_n=\ln(2n^2+1)+\ln(n^2+n+1)$ for $n=1,2,\ldots$. Prove that infinitely many $n$ satisfy $\{b_n\}\lt 1/2016$.
Here $\{x\}=x-\lfloor x\rfloor$ denotes the fractional part of $x$.
b) Let $b_n=\ln(2n^2+1)+\ln(n^2+n+1)$ for $n=1,2,\ldots$. Prove that infinitely many $n$ satisfy $\{b_n\}\lt 1/2016$.
Here $\{x\}=x-\lfloor x\rfloor$ denotes the fractional part of $x$.