1. Consider the sequence (an) with an = Vn2 + n - n, n = 1,2,3,,.......
Q2 (10 points) Vn2 + 4 – n, n E N. 2. Let (an) neN be the sequence with a, (a) Prove that lim,→0 an 0. lim,-00 bn, and prove the limit exists, by using the definition. (b) Let bn = n an . Find L =
1 (10 pts) Consider the sequence {anny where a, = tan' where a, = tan '- = 0. Prove that lim a, = 0 using our E-N definition of the limit of the nth term of a sequence. n n0
Question 2 (12 marks) (a) Consider the sequence with terms 2n35"5 log n n 1,2,3,.. 13 n8n (i) Determine whether ah diverges. If the sequence converges, find its converges or limit. o0 (ii) Determine whether r diverges. Justify your ansv swer an Converges o n-1 (b) Consider the series (2n)! 2 (n!) and determine whether it converges or diverges. Justify your answer IM8 8 Question 2 (12 marks) (a) Consider the sequence with terms 2n35"5 log n n 1,2,3,.. 13...
2. Exercise 2. Consider the sequence (xn)n≥1 defined by xn = Xn k=1 cos(k) k + n2 = cos(1) 1 + n2 + cos(2) 2 + n2 + · · · + cos(n) n + n2 . (a) Use the triangle inequality to prove that |xn| ≤ n 1 + n2 for all n ≥ 1. (b) Use (a) and the -definition of limit to show that limn→∞ xn = 0. Exercise 2. Consider the sequence (In)n> defined by cos(k)...
clean handwriting please Problem 1. Let {r,} be a sequence and L be a real number. Give the definition that lim, In L. Prove from the definition of the limit, that 2n2 + 1 lim nx 4n? - n + 1 %3D by completing the following steps. (a) Using the fact that 1 <n < n?, estimate from above the expression 2n? +1 4n2 – n+1 b) Given e > 0 find a threshold N, so that for all n...
5. Let {xn} and {yn} be sequences of real numbers such that x1 = 2 and y1 = 8 and for n = 1,2,3,··· x2nyn + xnyn2 x2n + yn2 xn+1 = x2 + y2 and yn+1 = x + y . nn nn (a) Prove that xn+1 − yn+1 = −(x3n − yn3 )(xn − yn) for all positive integers n. (xn +yn)(x2n +yn2) (b) Show that 0 < xn ≤ yn for all positive integers n. Hence, prove...
(1 point) Consider the sequence ax ncos(n) 2n-1 Write the first five terms of a,, and find liman. If the sequence diverges, enter"divergent" in the answer box for its limit. a) First five terms: b) lim,-- ..
Please solve #4 Solve problems below, Please show ALL your work! You will receive full credit only if you show all the appropriate steps. 1. In the problem below complete sentence in the definition of limit: Let (an) is a sequence. Number A is a limit of the sequence fan if for any 0 exists Ne such that Directly from this definition using e- N language prove that 1L lim -= n→oo n + 1000 3. cos n 5n2 +...
1. Let {n} be a sequence of non negative real numbers, and suppose that limnan = 0 and 11 + x2 + ... + In <oo. lim sup - n-00 Prove that the sequence x + x + ... + converges and determine its limit. Hint: Start by trying to determine lim supno Yn. What can you say about lim infn- Yn? 3 ) for all n Expanded Hint: First, show that given any e > 0 we have (...
real analysis. questions Prove that if lima In = 0 and > M for some M >0 and in 10 > 0, then lima (ny) - Asume 30 = 2,2-20+ In+1 = In + Prove that this sequence has a limit and find the limit. Prove that lim = L with L < if and only if every subsequence limo n L. Suppose that the sequence {an) is increasing and the sequence {yn) is decreasing. Moreover, lim a n -...