Question 178
a) Let $(a_n)$ be a sequence defined by $a_n = \ln(2n^2 + 1) - \ln(n^2 + n + 1)$ for all positive integers $n$. Prove that there are finite values of $n$ such that $\{a_n\} \lt \frac{1}{2}$.
b) Let $(b_n)$ be a sequence defined by $b_n = \ln(2n^2 + 1) + \ln(n^2 + n + 1)$ for all positive integers $n$. Prove that there are infinite values of $n$ such that $\{b_n\} \lt \frac{1}{2016}$.
b) Let $(b_n)$ be a sequence defined by $b_n = \ln(2n^2 + 1) + \ln(n^2 + n + 1)$ for all positive integers $n$. Prove that there are infinite values of $n$ such that $\{b_n\} \lt \frac{1}{2016}$.