1 Find L-! 2(s – 1)

Q: 2. A derangement is a permutation TE Sn with no fixed points, i.e. none of the elements remain in…

A: Introduction: Derangement is a very significant in combinatorics. Derangement is the process where…

Q: Let 8 = E--), Compute Sy for N = 1,2, 3, 4. Let S = n+2 n=1 Find S by showing that 1 3 SN = 2 1 N +1…

A: The nth term of the series is Tn=1n-1n+1.

Q: Show that the series Ln=l 12+2z10n-5z"ª –3z+2;10n converges absolutely for |리 < 1.

A: The series ∑n=1∞an is absolutely convergent if the series ∑n=1∞an  Ratio test:  The series ∑n=1∞an…

Q: Why is it that if I do row expansion on the first row, I get det(A) = -1, but, if I do it in the…

A: We need to find the determinant of a using the first row expansion and as well using the second…

Q: The roots of fourth order non linear homogeneous recurrence relation are 1, – 2,3 and - 4 with F (n)…

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Q: 2 Use Golden Section Search to determine (within an interval of 0.6) the optimal solution to igulum…

A: The function given is as follows: fx=x-ex We are asked to find the maximum of the function in the…

Q: Find f (n) when n = 4k, where f satisfies the recurrencerelation f (n) = 5f (n∕4) + 6n, with f (1) =…

A: According to the given information, It is required to find f(n) where:

Q: The formula for calculating the sum of all natural integers from 1 to n is well-known: n² + n 2 Sn =…

A: Given:  Sn= 1+2+3+.......................+n = n2+n2and,Qn=1.1+2.2+3.3+...............+n.n=n33+n22+n6

Q: 4. Find the expansion of (x + y)6 using Binomial Theorem.

A:

Q: 2. Prove that |µ(n)| _ S(s) S(2s) n=1 if s > 1.

A: Riemann zeta function is a complex-valued function defined as ςs=∑n=1∞1ns.  There is a relationship…

Q: 4. With the fact of 1² +2²+...+ n² = n(n+1)(2n + 1)/6, prove that n n n(n −1) ΣΣ(i-j)² 6 i=1 j=1…

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Q: The series 2" (n!)2 n=1 is divergent. Select one: True False

A: The given series is a 2n/(n!) 2. Repost the next question.

Q: (i) Give a definition of Fermat's Little Theorem. (ii) For all n e Z, show that 30| (n³ – n).

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Q: ii. Investigate convergence or divergence of 2n + 100 3n +1). n=2

A: Given series is  ∑n=2∞-1n2n+1003n+1n

Q: 4. Use the binomial theorem to find the first th ree terms in ascending powers of x of (1

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Q: Discuss the Convergence or Divergence for D { [4+ (-0"]" (Z+2)" こう 2n+1 - DZ² 2 لع Wi ngo (2n+¹)!

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Q: Find L {(2 - 3)}. 2(n!) 2 3 - sn+1 O 4(2n)! 12(n!) 9. - s2n+1 sn+1 O 4(n²)! sn2+1 12n! 9. - sn+1 O…

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Q: Use Heaviside expansion theorem (4s^2+7s+1)/s(s+1)^2

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Q: 6. Write the recurrence formula for Zj+1 - the derivative of second degree spline at node tj+1.

A: To give the recursive formula for zi+1- the derivative of second degree spline at the node ti+1

Q: If the increase in population at (n+1)th year of certain species on a culture is equals to the…

A:

Q: 3. Find the expansion of (x + y)$ using Binomial Theorem.

A: The solution is given as

Q: 7. Show that 1 En(ln n)P if and only if p> 1.

A: "Since you have asked multiple questions, we will solve the first question for you.  If you want any…

Q: 1. Prove that {2" : n E Z} = {,n :n E Z}.

A:

Q: –1)" Given: In(1 + x) = > . if a < 1. n + 1 n=0 (-1)" (n +2) 2n+1(n +1) 4. Compute the exact value…

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Q: 9- from the generating function, prove the recurrence relation: 2n In-1(x) + Jn+1(x): Jn(x) %3D

A: 9. Generating function is Jn(x)=∑r=0∞(-1)rx2n+2r1r!γ(n+r+1). We prove this in three steps. Step 1:…

Q: 7. If that second graph represents six games between four teams, and the score dif- ferences are…

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Q: Find Maclausin series of 4 fan ction Ź-22-3

A:

Q: Find f (n) when n = 2k, where f satisfies the recurrencerelation f (n) = 8f (n∕2) + n2 with f (1) =…

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Q: 2. For n > 0, let F(n) be the number of strings of length n over an alphabet of size k. Derive a…

A: Given: Let Fn be the number of strings of length n, for n≥0, over an alphabet of size k.   To…

Q: 2. Show that for every n 2 2 and every 0 <k<n- 2, within the triangle of Binomial Coefficients k+2…

A: We will use properties of binomial coefficient  nCr  +  nCr-1=n+1Cr+1Note : nCr=nr=n!r! n-r!   Proof…

Q: 1. Use method of undetermined coefficients to find the solution of non-homogeneous Fi- bonacci…

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Q: 2. Use the Binomial Theorem for the expansion of the function as a power ser: S(x) = V1+ 3x²

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Q: n>1 Problem 4. Prove by induction that, for every geq2 1 n +1 - - - 32 n2 2n

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Q: Determine the region of convergence for 100 3 n° + 2n2 +1 (z+1)" Σ 4n3 + 9 2n2 + 9 + Σ n = 101 3"…

A: Find radius of convergence, 1/R=an+1/an

Q: 1 is divergent. all n, then n=1

A: Given ∫0∞f(x)dx diverges∑n=1∞an convergent

Q: 3. (i) Give a definition of Fermat's Little Theorem. (ii) For all n E Z, show that 30 | (nº – n).

A:

Q: 5. Previously we used the recurrence relation dn+1 = 0.5dn + 200, d₁= 200 and the assumption that…

A: Given dn+1=0.5dn+200 and dn→L as n→∞. To find L .

Q: What is the sum of 1+1/3 +1/9+1/27+1/81 + ... if there is no sum, does it diverge?

A: Given that: What is the sum of 1+1/3 +1/9+1/27+1/81 + ... if there is no sum, does it diverge?

Q: 1. Show that 2n n=0 onverges absolutely.

A: Given series: According to the ratio test:

Q: 6. By Principle of Mathematical Induction prove 6n 6

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Q: 1c ical induction to prove 11

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Q: 7) Show that if a, > 0, and Lim na, +0, then the series aj + az + az + a4 + diverges.,

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Q: 4. Find the z transform of the sequence (2k} = {0,2,4,6,8, .} From ({k} (Izl > 1) (8-1)

A: In this question, concept of z-transform is applied. Z-Transform In its most basic form, the…

Q: 4. Prove that the L{2t²} = 4 S3

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Q: Let Z1,...,Zn be a sequence of random variables. For every n = 1, 2;..., P(Zn =n^2) = 1/n, and P(Zn…

A: Introduction:For a particular value of n, P (Zn = n2) = 1/n and P (Zn = 0) = 1 – 1/n.

Q: What is the solution to the recurrence relation: Tn=5nTn-1+6 with T0=3?

A: We will find out the required solution.

Q: 10. Find a recurrence relation satisfied by the sequence an = 2n2 + 3n + 1.

A: given,        a sequence an=2 n2+3n+1