9n – 5 a) lim n-- 3-n - 18 1 2n2 + 3n – 5 8n? +1 1 b) lim n- o0 || 4

Q: n(n+1)(2n+1) 2.Prove k2 = for n > 1. 6. k=1

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Q: n E(-1)* k + k)"-1, n2 1. k=0

A: The given value of ∑k=0n-1knkkm=0. The main objective is to detemine the value of ∑k=0n-1knk2+kn-1.…

Q: limn0 In(2n-3) -) is 8. n+5 Select one: 2 non of them

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Q: For all integers n > 1, E(4i – 3) = 2n² – n i=1 [ In other words: For all integers n > 1, 1+5+9+ ..+…

A: The given series is ∑i=1n4i-3=2n2-n, n≥1 To prove the statement by mathematical induction, we will…

Q: 13 e, thank you.

A: e)Given that, sum of the first n terms of the given series is

Q: 1. Prove that for any n > 2, n E N, one has (1-)(--)(--) - (-) - n +1 2n

A: I have solved question no. 1 according to policy rule.   To solve q.1, I have used method of…

Q: find the 4th term of (2n^2-3m)^4

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Q: 0.3.14: Prove 13 +23 + - +n³ = ()) (mla n(n+1) for all n E N.

A: Mathematical induction

Q: 2 4) The graphs of y=x and y=x make it obvious that x >x for all real number x > 1. Use induction…

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Q: THEOREM 4.1. Let l, k and t are both odd positive integers then for all B, a, a, b and c are…

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Q: [3n+ 2 (-1)A In 3n + 6 n=1

A: Express the alternative series test for convergence. If ∑n=1∞(-1)nan or ∑n=1∞(-1)n+1an converges if…

Q: 2. Let an = 3n + 2 for n = 0, 1, 2, ... (i ) Find ao, a1 and a2. (ii ) Show that a2 = 4a1 – 3ao.…

A: Answer and explanation is given below..

Q: 1. Complexity 2n – n³ + 2 Prove that O(n) 3n2 + 4n – 1

A: The big O notation is used to mention the important term r the biggest term in the equation. The…

Q: n+1 2n +3 lim 2n +1

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Q: THEOREM 4.2. Let l, k and t are both even positive integers then for all B, a, a, b and c are…

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Q: 2n+1+(-1)" Prove that for all integers n 2 0, 3.2"

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Q: 1 lim 1+ n 1 + 2 + n 1 +...+ n→∞ п +n, : =?

A: first, calculate the sum of the series then find the limit of the given problem as following:-

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

A: Given that bn=n3+n+65n3+4 and we are required to calculate limn→∞bn. Therefore, we are supposed to…

Q: (1.3) Show that n - n (n – k) . k + (k + 1) k k 1 - for any two positive integers k and n with k <…

A: Let's find.

Q: express     n-1 Σ((k^2)/n) k=1   in closed form.

A: The given question is related with closed form of finite sum. Given that,…

Q: Give a combinatorial proof of the identity: -1 k(²) = n(n=¹) k-1.

A: Use the following combinatorial formulae, to obtain the result. nr=n!r!n-r!n!=nn-1!

Q: 1 1 Let Hn = 1+ 2 1 +... + be the n-th Harmonic number. Prove that H2n <1+n for all nonnegative…

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Q: THEOREM 4.3. Let l, k are even and t is odd positive integers then for all B, a, a, b and c are…

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Q: 1 1 1 +- n-0n n+1 n+2 lim +....... 3n] equals

A: Given that : limn→01n+1n+1+1n+2+.........+13n This can be rewritten as,…

Q: (2) Prove that (1+x)" > 1+nx for all x > 0 and n > 1.

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Q: a) ) (tan¬'(n+ 1) – tan-1n Σ n=1 2n+1 b) 32n+1 n=1 2n +3 ) Σ 5n +1 n=1

A: To find whether the series converges or diverges and also their sum .

Q: a 1 In parts (a) - (c), prove that for all n E N: (a) n-1 k k=0 (b) ¿(:)-- 2n+1 k k-0 (c) E(-1)* k=0…

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Q: Prove n (2n = n n 1 k=1

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Q: 6. Verify Euler's Theorem for n = 16 and a = 5. (7 from 6.5)

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Q: Prove E-1(4i – 3) = n(2n – 1) for natural numbers n 2 1. i=1

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Q: Prove by induction  that the sum of the first n even integers starting at 2 is n(n+1).  (That is,…

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Q: 5. Algebraically show that n-1 k- = n · k k-1

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Q: 18.lim(3z + 2) = 3i + 2 of the following statement is not true? zi A O(3z + 2) – (3i + 2)| < E…

A: Option (B) true because

Q: 73. y" + 3xy' + 3y = 0. (-3)*x²47 y = a[1 + 2 (-34 x²] + a ₁ x + 2 ₁ 3.-5.-7. (2k + 1) ! ANS.…

A: Given y"+3xy'+3y=0

Q: 6* – 2 - lim x→0 ekx +7 k = (If any value of k will work, enter any; if none will work, enter none.)…

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Q: о Prove 2 (2n – 1) = n n - п - 1 k=1

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Q: Intro to proofs Given : AB=DC PROVE x=8 2x+5 =AB 5x-19= CD

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Q: Prove n 2 2n 1 k = n k - 1 n k=1

A: Binomial Theorem: According to the binomial theorem, let x and y are any two numbers. Then it is…

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

A: Divide by n^4 in denominator and numerator

Q: 5i In (2+ 5 lim

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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: 6. Using only Definition 4.1.: (a) For any real number k *(b) For any real number k 4n +1 (c) lim…

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Q: η Σ (1) 5 k=1 k odd = 6" - (-4)" 2

A: Here we prove that ∑k=1k oddnnk5k=6n--4n2

Q: 3n – 5n is n→ o n° - 2n² +1 lim 3

A: It is given that, limn→∞3n3-5nn3-2n2+1.We have to evaluate it.

Q: Prove that (-1+i3) + (-1 - if3)" has either the value 271 er the value -2" if n is ary n+1 Interer…

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Q: 7. Suppose a1, a2,...,a, are integers, r is a solution to the equation x" + a1x"-1 + a2x"-2 +... +…

A: for question number 7  we use contradiction to prove the statement.

Q: lim-(In(5n³-n+1)-in(7n³n+1)) 3 3

A: