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A sequence {an , is defined by the following formula. What is the limit of this...
Suppose that 20, 21, 22, ... is sequence defined as follows. do = 5,21 = 16,0 integers n > 2. Prove that an = 3.2" +2.5" for all integers n > 0. = 7an-1 – 10an-2 for all
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.)
2. (8 points) Let {fn}n>ı be a sequence of functions that are defined on R by fn(x):= e-nx. Does {{n}n>1 converge uniformly on [0, 1]? Does it converge uniformly on (a, 1) with 0 <a<1? Does it converge uniformly on (0, 1)?
3. (14 pts.) Let the sequence an be defined by ao = -2, a1 = 38 and an = 2an-1 + 15an-2 for all integers n > 2. Prove that for every integer n > 0, an = 4(5") + 2(-3)n+1.
We work with a sequence with a recursive formula is as follows, Xo = x1 = x2 = 1; In = In-2 + In-3, n > 3. The sequence therefore looks like: 1,1,1, 2, 2, 3, 4, 5, 7, 9, 12,... For example, x3 = x1 + x0 = 1+1 = 2, 24 = x2 + x1 = 2, and x5 = x3 + x2 = 3, X6 = x4 + x3 = 4, 27 = X5 + x4 =...
5. Evaluate the limit: lim expn n! n>00
4. Show that the sequence defined by a=2 An+1- 3-an satisfies () < an < 2 and is decreasing. Deduce that the sequence is convergent and find its limit.
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)...
Evaluate the following limit using Taylor series. 3 lim 2x2 zle x2 1 X>00
1. Consider the sequence defined recursively by ao = ], Ant1 = V4 an – An, n > 1. (a) Compute ai, a2, and a3. (b) For f(x) = V 4x – x, find all solutions of f(x) = x and list all intervals where: i. f(x) > x ii. f(x) < x iii. f(x) is increasing iv. f(x) is deceasing (c) Using induction, show that an € [0, 1] for all n. (d) Show that an is an increasing...