Problem 6) (The Cauchy condensation test] Let {an} be a nonincreasing sequence of positive numbers (an...
6. Suppose that {x,] is a sequence of positive numbers and limA = a Show that if L> 1 then lim x =00, and if L < 1 lim x = 0 n+02 b. Construct a sequence of positive numbers {x,} such that lim * = 1 and the sequence {x} diverges. c. Let k E N and a > 1 Show that lim = 0. O LIVE
Problem 2 Show that if the sequence of numbers (an)n-1 satisfies Inlan) < oo, then the series In ancos(nx) converges uniformly on [0, 27). This means, the partial sums Sn(x) = ) ancos(nx) define a sequence of functions {sn} = that converges uniformly on [0, 271]. Hint: First show that the sequence is Cauchy with respect to || · ||00.
(c) A sequence {2n} satisfying 0 < In < 1/n where E(-1)"In diverges.
Let (an) be a sequence such that lim an = 0. Define the sequence (AR) Exercise 21: by A =ļa, and An = zou-a + ax=a + zam for k21. Prove that an converges to some S if and only if Ax converges to S. N=0 k=0 Exercise 22: (Cauchy condensation test) Let (an) be a sequence such that 0 < antı san a) Show n=0 n=1 Hint: Recall the proof of convergence of for p > 1. Ren for...
Prove the ratio test . What does this tell you if exists? (Ratio test) If for all sufficiently large n and some r < 1, then converges absolutely; while if for all sufficiently large n, then diverges. lim |.1n+1/01 700 In+1/xn < We were unable to transcribe this image2x+1/2 > 1 We were unable to transcribe this image
Suppose a, and be are series with positive terms and on is known to be divergent. (a) If an > bn for all n, what can you say about a,? Why? o a converges by the Comparison Test. o s a diverges by the Comparison Test. Ο We cannot say anything about Σας: o a, converges if and only if n-a, 2 bn O a converges if and only if 2a, 2bn. a? Why? (b) If an <bn for all...
2. Suppose that (an), İs a sequence of complex numbers such that there exists a positive number 0 such that for all NEN an M (i) Show that (ON)N converges to a number . (ii) Show that sx -2Nan for N E N is a Cauchy sequence 2. Suppose that (an), İs a sequence of complex numbers such that there exists a positive number 0 such that for all NEN an M (i) Show that (ON)N converges to a number...
Assume that the sequence defined by a1 = 3 an+1 = 15-2·an is decreasing and an > O for all n. Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.)
Given that the sequence defined by - 1 2+1 = 5-1 an is increasing and an < 5 for all n. Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.)
The sequence (Un) of positive real numbers satisfies the relationship In-1XnXn+1 = 1 for all n > 2. If x1 = 1 and x2 = 2, what are the values of the next few terms? What can you say about the sequence? What happens for other starting values? The sequence (yn) satisfies the relationship Yn-1Yn+1 + Yn = 1 for all n > 2. If y1 = 1 and Y2 = 2, what are the values of the next few...