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=na_(n−1), with initial term a0=1.Use iteration to find a closed formula for the sequence. Simplify your answer whenever possible.

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.

Suppose ao, a1, a2,... is a sequence such that ao = a1 = 1 and forn2 1, an+1 = n(an+an-1). Find az = a5 = -, a7 =

Let {an} be a monotonic sequence such that an ≤ 1. Discuss the convergence of {an}. When {an} converges, what can you conclude about its limit?

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

Let {an} be a real number sequence. Prove that                             {an}= {1+(1/2n)}  -------> 1

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

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