6 let n 1 3 2n 20 greater than 0 hint if a b greater than 0 then a les

Question

6. Let n+1 = √3+2n, 20 > 0.
Hint: If a, b>0 then √a<√ba<b.
(a) Find the fixed point L for which n+1 = n = L.
(b) Show algebraically that {n} is increasing, i.e. In ≤In+1 when In € [0, L].
(c) For which zo is {n} bounded above by L?
(d) What does the monotone convergence theorem tell you about limn→ In?
(e) Write a code to approximate limon when zo = 1 with N = 20 terms.
(f) Show that the convergence (sequential rate of convergence) is [at least] linear.