Prove that the sequence (xn)n≥1 converges and compute limn→∞ xn

Define (xn)n≥1 by x1 := 0 and xn+1 := (xn + 3)/4 for n ≥ 1. Prove that the sequence (xn)n≥1 isconvergent and calculate the limit limn→∞ xn

Let f : (0, ∞) → R be a decreasing function. A sequence (xn) is definedby x1 = 1 and xn+1 = xn + f (xn). Show that limn→∞ xn = ∞.

Define (xn)n≥1 by x1 := 1 and xn+1 := (3xn + 3)/4 for n ≥ 1. Prove thatlimn→∞ xn = 3.(Hint: First show that xn ≤ 3 for all n ≥ 1.

Prove from first principles that the sequence (an) is convergent, wherean = (−1)n + n2n + 1for all n

Prove thatlimn→∞(4n)1/n = 1