Suppose that the sequence {an} converges to a and that la| < 1. Prove that the sequence {(an)"} converges to 0.

Prove that the sequence {cn} converges to c if and only if the sequence {cn- c} converges to 0.

Let (a_n) be the sequence defined by the recursive relation a_n=a_(n−1)+5n+1, with initial term a0=0.Use iteration to find a closed formula for the sequence. Simplify your answer whenever possible.

Write out the first five terms of the sequence , determine whether the sequence converges, and if so find its limit. [(-1)*-1 ¯ (n+4) . n=1 Enter the following information for an = (-1)"-1 (n+4)² · = Ip a2 = аз a4 A5 = (-1)"–1 lim (n + 4)² Does the sequence converge

a) {(SN +17)/(2N)} 10. Prove that the sequence {(-1)"} does not converge. 11. (r.} be a

PLEASE DO NOT prove by contradiction. Thank you very much!

2) Show that the sequence defined recursively by 3 4 An aj = 5 аn+1 satisfies 2 < an < 5 and is decreasing. Show that the sequence is convergent and find its limit.

Find a sequence {an} whose first five terms are                                                 2/1 , 4/3 , 8/5 , 16/7 , 32/9 , . . . and then determine whether the sequence you have chosen converges or diverges

Suppose that the sequence {an} is monotone and that it has a convergent subsequence. Show that {an} converges.

Suppose that {xn} is a bounded sequence such that all convergent subsequences converge to the same numbera. Prove that {xn} must converge to a.