Show that 100 1 arctan(100) – arctan(2) + Σ 1 < arctan(100) – arctan(2) + 1+ 22 1+ 1002 1+ k2 k=2 WI

Q: In the domain of Gaussian integers Z[i], the element 23 is * Irreducible Reducible Unit None of the…

A: (i) Every prime is irreducible in Z[i].

Q: The Hermite polynomials Hn(x) are special functions that are extremely im- portant in quantum…

A: For first proof differentiate exponential function and for second part Differentiating series

Q: Example 7-27. Let X1, X2, ..., X, be jointly normal with E (X;) = 0 and E(X?) = 1 for all i and Cov…

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Q: () = 2 sec(a O conon1 O pton 2

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Q: X sec10 can 1. Let u = 7 In х2 - 1

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

Q: shor w that To- {0} abelian 7oup. where p an is prime.

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Q: @) 2) Fs of (x) = -PFC (Psy) -} [ If f (x,y) → as 6004X proof. (H-w.) then в

A: Introduction: The Fourier sine and cosine transforms are variations of the Fourier rework that do…

Q: 1. Using (2n +1)Pn= Pn+1-P-1: prove that P(x) = (2n-1) Pn-1(x) + (2n-5) Pn-2+(2n-9) Pn-3+.... - and…

A: Given : 2n+1Pn=Pn+1'-Pn-1' To Prove : (a) Pn'x=2n-1Pn-1x+2n-5Pn-2+2n-9Pn-3+ . . . . . .…

Q: 2. Let ACR. If a is a lower bound of A, then it follows that -a is an upper bound for A.

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Q: Show that >n"x" diverges for all x + 0. n=0

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Q: Assume that after w months,butterfiles populace in 1000 is calculated as F(w)=12w+cos(16w)+n×100…

A: Minimum and maximum of a function depends upon all the variables present in the function.

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: х Prove that 1 < < sin x for all x e (0,5). - ド|N

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Q: 3 3 Prove that 2. j=l k=1 Ey =1 if ijk = 123, 231, or 312 Eyk -1 if ijk = 321, 132, or 213 Eyk =0 if…

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

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Q: 우(4,3) - +xy Elewhere a fivd 우 (s) 6) find p( ys%2 /x= 0:3)

A: A joint probability distribution gives the probability for two variables together which is…

Q: 1. Find the Wronskian for yi = x , y2 = x² and y3 = x³.

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Q: How does Abel's Identity allow us to conclude that if the Wronskian is 0 at a point t, then the…

A: To prove: By Abel's identity conclude that if Wronskian is 0 at a point t, then the Wronskian is…

Q: 15. Let z = 2cos0 + i(1 – 2sin0) where -n < 0 < n Show that Iz – il = 2

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Q: X1,, Xn ~ N(-3,2) (random variables) ... Z, = 1(X,X2 + X1X3 + X1X4 + •..+ X1X„), n > 3 %3D (a) Find…

A: From the given information, X1, .....,Xn~N-3,2 And Zn=1nX1X2+X1X3+...+X1Xn, n≥3 Consider,…

Q: 1. Show that sinz is unbounded on C and lyl <|sinz] <ebl z=x+yi € C. for

A: Given:         We have to show that sinz is unbounded in ℂ and y≤sinz≤ey   for z=x+yi ∈ℂ Function…

Q: Show that the series n=0 xn cos(nx) x² +n³ + 1 defines a differentiable function g(x) for x in [-1,…

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Q: 12. Start with the identities x2 1 P, (x) = xPn-1(x) + *P(x) n P, (x) = xP-1(x) + nP,-1(x ) (a) Show…

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Q: 1:if z 1 then z is even " is false, we need to find z such that

A: Given statement is

Q: Let 3 2 1. Verify that I4 = A. Show all of your work. 2. Verify that Alz = A. Show all of your work.

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Q: 7. (a) Find all the values of z such that e = 1+ i.

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Q: Example 6: State the third congruence that is needed to prove AOZN = AELD by the indicated method. W…

A: Given,

Q: Find the limit of the sequence bn = (cos n)/(n2 + 1).

A: Given sequence is bn=cosnn2+1.

Q: two positive iontages and bare by 2, them prove that. (2° +1) is most divisible by (2²-1) let a with

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Q: 2. (a) For n ≥ 0 and x E R, prove that (b) Prove n n Σ ^ (^.) 2^* (1 - 2)"-* k k=0 = = In. n x)n-k…

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Q: O If z-frxy) show that つる

A: As per our guidelines we are supposed to answer only one question. For the solution of the problem…

Q: 1. Iff: [0, 1] R is defined as f(x) = x if x is rational, 1-x if x is irrational, discuss the…

A: Given :  f: [0, 1] →ℝ  is defined as     f(x) = x,if x is rational1-x,if x is irrational To find :…

Q: 1 find func- 88. Finding a Composition If T(x) = %3D Vi + Vx tions f. g, andh such that fo goh = T.

A: Given: T(x)=11+x To find f, g, and h such that f∘g∘h=T

Q: 2.2.6. In the text, the modulus is defined in terms of the inner product. Prove that this can be…

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Q: 3 5 The Wronskian of {X2,x2}

A: The given set is: x32,x52

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

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

Q: Suppose an 20 and limnan = L.Then 1. If 0 ≤ L < 1, then Enz1an converges

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Q: 5. Prove that . Hint: Compute e-' from the continued frac e - 1 1 + 2 + 3 + ·…. tion for et given in…

A: When a series is in the form ∑k=1∞-1k-1α1α2α3...αk, we can express it as a continued fraction as…

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: Let yII + yI - 42y = 0 Find all values of k such that y = ekx

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Q: If ||<1, determine the maximum modulus f(=) =+1 (a) f(=)==-2z+3, (b) f(=)==+=-1, (c) f(=)= 2z-1' (d)…

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Q: 5b) Let f (x, y(x)) = y(x) + 1 and y(0) = yo = 2. Use Picard-iteration and induction to show that YN…

A: Please go through the solution.

Q: 17. Prove thatsin x = =x 1-x2 dx

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Q: 4. Letf(x)=x22/5 - 19x². Determine the largest n for which fis n-times continuously differentiable.

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Q: consider fixi) = %3D show that trany2 eloes net suhsty kipschifzconchition elo muiswechenclndes ino…

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Q: 1) Prove that 2" 4 2) Prove that 4 divides 7" -3" for eachn E N 3) Prove that | sin na| 3 f(n) =…

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Q: Theorem 26. If n ∈ N then (1^n) = 1. Theorem 27. If x, n, m ∈ N then (x^m)^n = x^(mn) . Exercise…

A: Given 1n =1 where n belongs to N(natural numbers)   Prove .. 1×1×1×1.........×1 n Times.........(1)…

Q: Suppose the interval [1,9] is divided into n subintervals of equal width, and Ck represents the left…

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Q: Let A={(-1)" Find sup A and inf A. Provide some explanations how you got their values or why they…

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Q: Let 0 < 0 < 1 be a rational number. Show that the sequence {cos(nT0)}-1 is divergent.

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