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.)
Show that a bounded and monotone sequence converges. Here a
sequence
is called monotone, if it is either monotone increasing, that is
for all
or monotone decreasing, in which case
for all
.
in Sn=1 An+1 > an neN an+1 < an We were unable to transcribe this image