2. Prove that lim (-1)"+1 0. 72-00 n 2n 3. Prove that lim noon + 1 2. 80 4. Prove that lim n-+v5n 0. -7 9 - in 5. Prove that lim n0 8 + 13n 13
8-7. Find the smallest positive integer a such that 5:13 +13n" + a(9) = 0 (mod 65) for all integers n.
Please only do number 8.
7. Prove that if {zn} is a sequence with lim zn w and lim zn = w, then lim z 8. For the sequence of the previous exercise, prove that lim 1/zn = 1/w provided w 0.
m2 2. Prove that lim -+0n3 + 1 -=0. 3 5 100 3n2 + 2n - 1 3. Prove that lim = 5n2 +8 cos(n) 4. Prove that lim = 0. n-700 m2 + 17 5. Prove that lim (Vn+1 - Vn) = 0 Hint: Multiply Vn+1-vñ by 1 in a useful way. In particular, multiply Vn+1-17 by Vn+1+vn
3. Find: 7T (1) lim n sin (3) lim arcsin G n-00 n 100 COS- n n00 n 1 (4) lim (1+ (6) lim (n+1 n) n- 3n n-00 1/n (2) lim arctann Vn2 - 1 (5) lim 2n In (1+) (8) lim n (11) lim n+ n 2+1 (14) lim n- 75n+2 (9) lim (nt n-00 700 n->00 n (7) lim V (Vn+1- Vn) (-1)"n (10) lim n+ on+1 (13) lim (3" +5")1/n sinn (12) lim arctan 2n 2n...
ad cal 2
. Problem 8, (10 pts.) Prove that on the interval [0,0.8) -2n lim dx Problem 9, (10 pts.) na(1-z)". Let fn (z) Prove that . Problem 10, (10 pts.) Using Method of mathematical induction prove that: If function u(x) is such that a,--u then a ,u u, 2n1
. Problem 8, (10 pts.) Prove that on the interval [0,0.8) -2n lim dx Problem 9, (10 pts.) na(1-z)". Let fn (z) Prove that . Problem 10, (10 pts.)...
2. Prove that lim (-1)"+1 = 0. n-00 n
3. (10 marks) Find the limit and prove it using the definition. 4x2 + 13 lim x+ x2 + x + 1 4. (10 marks) Find the limit and prove it using the definition. 4x3 + 13 lim *40x2 + x + 1
2n 3. Prove that lim n+on+ 1 2.
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Honey
9 6 14 14 15 12 13 14 7 10 11 14 9 12 13 13 5 5 9 6 7 8 14 7 13
DM 5 5 12 6 11 7 13 7 7 5 6 5 9 5 13 11 13 7 10 8
control 2 4 4 3 1 9 7 4 9 8 2 8 6 4 2 5
Calculate the mean, median and mode for each of the data
(Honey, DM and Control)....