Now, in in lim n → o n i = 1 30i -9 + lim n → o n 30 -9 + = 1 i = 1 But s -9= and i = 1 i = 1

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Q: 1. Assume that cin. 8" + c2n · 4" + c3n° . 9" = 0 %3D for all n> 0. Show that then ci = 0.

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Q: E an diverges and lim an/bn = 0, then b, diverges. n=1 If n00 n=1 True False

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Q: 10.1 In(n+1) an= (In(ne) ntl L.H. In(@+1) In) lim In(n+)) Vn lim 00 lim スn lim lim lim %3D %3D (n+1)…

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Q: 12. Given: L1 = Z2; Z3 = 4 Zs 2 and 3 are complementary Prove: Ls 1 and 4 are complementary 2,

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Q: Let bn : n° +n+6 for all n ɛ N. Find limn→∞ bn- 5n3+4

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Q: Use the identity 1/(k(k + 1)) = 1/k – 1/(k + 1) and Ex- ercise 35 to compute E- 1/(k(k + 1)).

A:

Q: Prove that 2n3 + 5n 1 n? (i) lim n→∞ 4n³ + n² (іi) lim 1. (п — 7)2 — 6 2 n-00 ||

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Q: 11. Show that ((n + 1)!)" < 2! · 4! · (2n)!. 12. Show using induction that for n e N, n 1 n i·(i+1)…

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Q: 27 en+1 – 1 < 0.05 ? 2 (n + 1) · (n + 1)!

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Q: 2. Prove by induction that n n(n + 1)(2n + 1) lim ? 6. Evolueto the not oron hotwoon the curvo flo)…

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Q: 17. prove that Edln H(n/d)7 and σ(n ) σ(n) Σ %3D

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Q: lim n->infinty ((4n+4)(4n+3)(4n+2)(4n+1))/((n+1)4 3964)

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Q: 15. Prove the following identity C) + (" ; ') + (" * ) + (" ; ) +-+ " : )--- " * ** ') (n + k + k…

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Q: 2. (a) Evaluate 7n+1 +2" i. lim N; n-00 7" – 2n | ii. lim Vn2 + 3n – Vn2 – 3n |, n € N, n > 3. n-00

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Q: 2. If L, and S, is its n-th partial sum, find: 3 (a) lim cos(an) (b) lim S

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Q: For any integer n2 1. prove that: (a) 1+2+3+..+ (n – 1) + n + (n – 1) + . .. +3+2+ 1 = n². (b) +...+…

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Q: a. b. In n {an} = {1 - (n In (n+1) 1 8n²16n +6 7=200 + T- en-

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Q: lim-(In(5n³-n+1)-in(7n³n+1)) 3 3

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.