Sequences and Limits

Consider the real sequence {x_n}_n generated by the iteration scheme x_n+1 = x_n(2-ax_n), for n = 0, 1, 2, ...... where a>0 and x_0 satisfying 0 < x_0 /=x_n>0 for all n. b. Prove x_n>/=x_n-1. c. Conclude that lim n-->infinity ...continues

Limit Proofs

Prove the following a) If lim n-->infinity (a_n*b_n) exists and lim n--> infinity (a_n) exists, then lim n -->infinity (b_n) exists. b) If lim n--> infinity (a_n) = 0 and {b_n} is bounded, then lim n-->infinity (a_n*b_n) exists and equals 0. c) If lim superior (a_n) exists, then {a_n}_n is bounded above.

Cauchy Sequence and Limit Supremum

Suppose that {a_n}_n is a real Cauchy sequence. Prove that lim superior n--> infinity (a_n) = lim inferior n--infinity (a_n) so as to conclude that lim n--infinity (a_n) exists.

Proofs : Bounded Sequences

Suppose the sequences {a_n}_n and {b_n}_n are both bounded above. a) Prove that for all n in the naturals sup{a_k + b_k: k>/=n} is less than or equals sup{a_k:k>/=n} + sup{b_k:k>/=n} b) Use this to conclude: limsup (a_n + b_n) is less than or equals limsup(a_n) + limsup(b_n) (all limits are n--> infinity)

Proof : Sequences and Supremum

Suppose that the sequences {a_n}_n is bounded above and lim(b_n) exists. a) Prove that for all e>0 there is an N st that for all n>=N sup{a_k:k>=n} + b_n <=sup{a_k + b_k: k>=n} + e. b) Use this to conclude limsup(a_n) + lim (b_n) <= limsup (a_n+b_n) (all limits are n ---> infinit ...continues

Series : Absolute Convergence

Suppose the summation from k=1 to n of a_k is absolutely convergent and {b_n} is bounded. Prove that this implies the summation from k=1 to n of a_k*b_k is absolutely convergent.

Cauchy Sequences

Consider the real number iteration scheme x_n+1 = f(x_n) for n = 1, 2, ... with x_1 given. In addition, suppose there is a number 0 < p < 1 st lf(x) - f(y)l < = plx-yl for all x,y. a) Show lx_n+1 - x_nl < = p^n-1lx_2 - x_1l for all n. b) From this, conclude {x_n}_n is Cauchy.

Proof : Sequence of Partial Sums

Suppose a_k is a nonincreasing sequence satisfying a_k --> 0 as k--> infinity. Also suppose the sequence of partial sums by s_n = l summation k = 1 to n of b_kl is bounded. Show that these conditions imply summation k = 1 to n of a_k*b_k is convergent.

Financial Accounting

Leases R Us, Inc. has been contracted by Robotics of Beverly Hills to provide a machine that would assist in automating a large part of their current assembly line. Annual lease payments will start at the begining of each year. The purchase price of this machine is $200,000 and it will be leased by RBH for 5 years. LRU will util ...continues

Bounded Linear Operator : Bounded Invertible an Norm

11.8 Let and where a "1" appears in the n-th position and a zero in all other positions. Let (an) be a sequence of complex numbers. Prove then that (i) ... defines a bounded linear operator on G if and only if... , and accordingly find the norm of T. (ii) What are the necessary and sufficient conditions for T to ...continues