Question 2.3. Prove the following: if lim sn = L, and tn = 5n+1, then limt,...
(b) Show that if lim sn oo and lim tn > -oo, then lim(sn + tn) 1 +oo.
Please answer all parts. (2) (a) Give an example of sequences (sn) and (tn) such that lim sn ntoo 0, but the sequence (sntn) does not converge does not converge.) (b) Let (sn) and (tn) be sequences such that lim sn (Prove that it O and (tn пH00 is a bounded sequence. Show that (sntn) must converge to 0. 1 increasing subsequence of it (b) Find a decreasing subsequence of it (3) Consider the sequence an COS (а) Find an...
6. Let si = 4 and sn +1 (sn +-) for n > 0. Prove lim n→oo sn exists and find limn-oo Sn. (Hint: First use induction to show sn 2 2 and the.show (sn) is decreasing)
IDY in < oo and lim - Yn < 0o. Prove that lim,+ 1. Let In > 0. Yn > 0 such that lim,- Yn) < lim,-- In lim,+ Yn: i tn < oo and lim yn < . Prove that lim. In 1. Let In 20, yn 0 such that lim Yn) < limn+In lim + Yr
Question 1 The series n²tn n=1 73/2+5n+1 converges. O True O False
1. It is known that has a Fversion of the test. Find the limt lim (Vx+&+1 -2 +2 -1 ) Show your work in the PDF version of the best. Done 2 of 13 10:55 1 of 3 CO 2020 VIPadron
(Exercise 4.13, reordered) Given a series ΣΧί ak, let 8,-Ση-i ak. Σχί ak is Cesaro summable if S1 + 82 +... +Sn lim n-+o converges. (a) Give an example of a series Σ00i ak that is Cesaro sum mable but not convergent (b) Prove that if 1 ak converges, then it is Cèsaro summable. Hint: Say the sequence of partial sums sn → L. Try to prove that =1 8k → L by showing and then splitting the latter sum...
(3) Use the definition of convergence to prove each of the following (a) 1 is not the limit of the sequence sn (-1)" (b) lim = 1/2 2n (c) Suppose that lim an = a. Prove that lim 3 . an За.
1. Find 7(5") 3" 5n+12n+1 sin(n!) lim (a) im (b) n 2018 пH0 п2 n oo _ Зп? + п + 4(3") nt T lim sin (d) (с) lim 1 - 2n 3п — 2п n oo n oo 2. Let (an), be a sequence such that lim an n oo L. = 1 (a) What is lm an+1? пH00