Question 1
For each positive integer $n$, consider the polynomial $P_n(x)=x^{3n}-3\cdot4^{n-1}x^n-2^{3n-3}$.
(a) Prove that $P_n(x)$ has exactly one positive real root, denoted by $a_n$.
(b) Put $b_n=(2-a_n)/n$ and $c_n=b_1+b_2+\cdots+b_n$. Prove that the sequence $(c_n)_{n=1}^{\infty}$ has a finite limit.
(a) Prove that $P_n(x)$ has exactly one positive real root, denoted by $a_n$.
(b) Put $b_n=(2-a_n)/n$ and $c_n=b_1+b_2+\cdots+b_n$. Prove that the sequence $(c_n)_{n=1}^{\infty}$ has a finite limit.