[Solved] MA3831 Homework 2

Let a_n be a sequence which converges to a positive number A. We showed in class that there is an N in N such that for all . From there, show that converges to .

exercise 2:

Optional 2.6.G from Davidson Donsig.

exercise 3:

Prove or disprove: Let a_n be a sequence of real numbers. If lim (a_n+1 a_n) = 0, then a_n is convergent. n

exercise 4:

Let q be a fixed positive number. Show that the sequence is eventually decreasing.

exercise 5:

2.6.B from Davidson Donsig. Hint: set f(x) = 5 + 2x. Solve f(x) = x and the inequality x f(x). Prove by induction that 0 a_n a_n+1 1 + 6.

exercise 6: 2.7.A

exercise 7:

2.7.G. Hint: any integer p can written as 3n 1, 3n, or 3n + 1.