Prove that L ={(1Z*)"10" | n e N}is not regular.

Q: Prove that Eo 2i = 2n+1 – 1 - i3D0

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Q: 1-z+2 dr +10+9 Use the Residue theorem to evaluate

A: The integral is I=∫-∞∞x2-x+2x4+10x2+9dx. Consider fz=z2-z+2z4+10z2+9 By theorem, if a > 1 and M…

Q: 10. Show that for all n E Z, n² – 3n +1 2.

A: Use mathematical induction to prove the given statement. Check whether the result is true for n=1.…

Q: 1 se mathematical induction to prove that 1 + + > Vn +1 for every integer n > 3. + ... Vn

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Q: 4. Prove that for n > 0: E, 2* = 2"+1 – 1. sk3D0

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Q: s expansion of 2 , valid in the re of 1/z2 is:

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Q: X sec10 can 1. Let u = 7 In х2 - 1

A: Given: ∫x sec107ln2x2 - 13tan7ln2x2 - 13dxx2 - 1.

Q: Prove using the IVT. For all positive integers k, cos x = xk has a solution.

A: Intermediate value theorem: Let f be a function continuous on a closed interval a,b. Let k be number…

Q: Evaluate the integral. 2 (3u - 5)(u + 2) du -1

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Q: 6. Find a²z дгу of z = In(xy)

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Q: 1 dæ using the residue theorem. (2² + 1)° Evaluate the improper intergal -00

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Q: 3- Vn n=1

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Q: Prove H'n(x) = 2nHn−1(x), n = 0, 1, . . .

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Q: Prove that %3D 3 and hence. deduce that. 3--(n+3)Jn43+ (n+s)Jnes to 2.

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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: (9) Is an complex annullus Ann(2o, R1, R2) with 0 < R1 < R2 simple connected?

A: Note: As per bartleby instruction when more then one question is given only one has to be answered.…

Q: Prove that 1/(nx) → 0 pointwise, but not uniformly on (0,1) as n → co.

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Q: 3V Show that there dã do not exist two real numnbers which add to 7 and multiply to 13. It is not…

A: our objective is to conclude the result

Q: 7. Find a pair of integers (x,y) such that 4021x + 2014y= 1.

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Q: Find the order of [5] in Z23 using Fermat's Theorem.

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Q: To prove that dx In /1+ (In x)² + In x + C x/1+ (In x)2 we use the substitutions

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Q: 1. Find an r> 0 such that all zeroes of 24 + 6z²+z+3 are contained in D,(0).

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Q: Show that yn for all n=1, 2₁- ✓

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Q: Prove that z'e'-z = 1 has exactly two roots inside the circle |z] = 1.

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Q: state and prove the Cauchy's Residue Theorem?

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Q: Compute 32+n21-3n n=0 If this is divergent, input "O" as your answer If applicable, submit answers…

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Q: 5.Z-transform of {u(n) + 2"- nu(n) } is

A: Note:- As per our guidelines we are supposed to answer only one Question as this is multiple…

Q: iii) a) Prove that Pa(-x) (-1)".Pn(x) u (2n)! 22 (n!) b) Prove that P2n(0) = (-1)".- %3D

A: To prove- Pn(-x)=(-1)n.Pn(x)P2n(o)=(-1)n.(2n)!22n(n!)2

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: 7. Prove that 2. O = (-1). %3D

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Q: A cubic spline S is defined by: (So(x)%D 1+x- x (5,(x) = 2 - 5(x – 1) + (x – 2) if 0 SxS1 if 15xS2…

A: As we all know that Natural Cubic Spline is a piece-wise cubic polynomial that is twice continuously…

Q: 2 (x) 2. prove that s

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Q: 5) Prove that d tanh ) |-x2

A: Prove that  d/dx[tanh-1x] =1/(1-x2)

Q: Find the residue of 3 Z 2 z² - 1 at z→ ∞0.

A: We have to find the solution of the given problem.

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

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Q: 5. Use Cauchy's Residue theorem to evaluate dz; Có lz-il =1 (2+1)

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Q: 1. Prove by induction that if y = , then d"y = (-1)"n!x¯n-1, dx" where n! = 1 · 2..... n is the…

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Q: LET X 2 An be a SEQUENCE OF COMPLEX NUMBERS THAT CONVERGES TO ZERO LIM X=0 PROVE THAT THE OPERATOR…

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

A: Given series: According to the ratio test:

Q: Use Euler-Maclaurin's formula to prove that E n² = "(n+ 1)(n +2) 6. %3D 1

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Q: Prove that 2n+l > (n + 2) · sin(n) for all positive integers n.

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Q: IRI- 2. <| sinz|< -e 2

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Q: Prove that f(n) = 1000n5 + 20000n2 + 32 is O(n6 ); Is this a tight upper bound? Why or why not?

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Q: Q1. Prove that KT = = -.

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Q: For n2 1. prove using mathematical induction: 2" s 2n+1 - 2n-1 - 1

A: We have to solve given problem:

Q: 4. Prove that the L{2t²} = 4 S3

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Q: 3. Prove that the sandwitch theorem.

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