2n 3. Prove that lim n+on+ 1 2.
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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...
3) Determine whether the given series converges or diverges. i) (7 pts.] 4n2 + 3η +1 2n2 - η n=2 ii) (8 pts.] Σ 2n 11 COS NI iii) (10 pts.) ΠΕ1
(2) Prove by induction that for all integers n > 2. Hint: 2n-1-2n2,
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
2n2-31+1 b. lim cos 1. Find the following limits: a. lim 1 00 --- 3n2+4 n-00
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...
QUESTION 5 (a) From first principles ( ← N ) prove that inn an = 5 where n5n-1 (in) = (n2-7 + n + 5 (b) Use an 6 argument to show that r22 2 lim
Exam 1, Q1: 10 (a) (77-5V5) 33 10 10n 4i lim 10 (b) 7-5) nco n 2n 3 5 (c) (7-55) 5 (d) (7-5) 2 (e) (7-5) 2 A W
Exam 1, Q1: 10 (a) (77-5V5) 33 10 10n 4i lim 10 (b) 7-5) nco n 2n 3 5 (c) (7-55) 5 (d) (7-5) 2 (e) (7-5) 2 A W
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