Derive a formula for the nth derivative

Question

Solve the following:

Q1. Determine whether....converge or diverge

(i) ^n=∞_n=1Σn+1/3n²+5n+2

(ii) n=∞_n=1Σn^n/n!

(iii) n=∞_n=1Σx^n/n²
where x∈R

Q2. Given that a_n,a_n-1,.....a₁,a₀ are all integers and that an and a0 are non zero,derive a necessary condition for the equation

a_nxⁿ+a_n-1x^n-1+....+a₁x+a₀=0 to have a rational root.Then use this condition to prove that √n-1+√n+1 is irrational for every integer n>=1.

Q3. Using binomial coefficient,derive a formula for the nth derivative of the product of two function.

Q4. Suppose that f(x)has a continous first derivative for all x∈R.

(a) Prove that f(x) is concave if and onloy if f(x*)+(x-x*)f'(x8)>=f(x)for all x and x*R

(b) Given that f(x) is concave,prove that x*is a global maximum of f(x) if and only if f'(x*)=0.

(c) Given that f(x) is strictly concave,prove that it cannot possess more than one global maximum