(5) Use the recurrence for the Catalan numbers Cn+1 ating function Ek=o CkCn-k to prove that the gener- f (x) = Cna" %3D n=0 satisfies the equation 1- f(x) + xf(x)² = 0. (This implies that f(x) = 1-V-4=). Hint: The coefficient of x" in (Eo Chak )² is E-o CkCn-k· –4x k=0

Q: Prove that Ją (2) = =- J¼ (2) – J_4 (=)

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Q: Q2] Find the lower bound of Iz - 3|if |z - il < 1

A:

Q: Evaluate z' if z = [1 + (x² − 1)³] ²³ O a. 2(2²-1)² (1+(2²_1)²³) i O b. none of these О с 41(2²-1)²…

A:

Q: if f(x1,x2) = e-*-*2, 0 < x1 < o, 0 < x2 < o, F(x1, x2) = e-Xie-x2 + e-X2 True False

A:

Q: -35 Find s?(s°+4)

A: We will find out the required value.

Q: (b). Use the definition of the derivative to evaluate f (x)=x". [Hint: Use the Binomial Theorem:…

A: Given: i. fx=xnii. a+bn=an+nan-1b+nn-12!an-2b2+·····+nabn-1+bn. To Find: Use definition of…

Q: (4) VL1, L2 ((L1 UL2)* = L₁* U L₂¹).

A: This is always True

Q: Find the coefficient of x10 in (1 +x+x²+x³ + •..)" Find the coefficient of x' in (x +x6+x² + • ·…

A: 1)Given expression1+x+x2+x3+....nThe series 1+x+x2+x3+.... follows a geometric progression with…

Q: The solutions of cosh(2x) = 8sinhx - 5 are a) x = ln (4+ √17) and x = ln (1 + in (1 + √2) b) x = ln…

A:

Q: to Prove that : Ge = { in (+ + ² ) Z tan z =

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Q: 0. Reciprocal Substitution (x´ - x')"3

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Q: In x /1+ (In x)² -dp-

A:

Q: Specify c-1,Co, c₁ and p such that the formula f'(x) h (c-if(x-h)+cof(x)+c₁f(x+h)) denotes the…

A: Here , f'x≈h-pc-1fx-h+c0fx+c1fx+h. To Find: The value of c-1, c0, c1,p.

Q: 5.16] Use the function f (x) = xe¯ ¯x² to find fave on [0,4] %3D

A:

Q: Evaluate by using definition Gamma Function 2

A:

Q: Let F(x) = f(x) = x" for x = [a, b] and some positive odd integer n. Show [₁ 62_a²n 2 fdF =

A:

Q: Show that when |+ 1|<1, = =1+E(n+1)(=+1)"

A:

Q: (4) (LATEX question) (2018A8(b)) Use the Intermediate Value Theorem to prove that for each real…

A:

Q: 25 (6x + 5)³ But Do NOT use the Quotient Rule. a) Let f(x) = = f'(x) = Find f'(x). b) Let g(x) = 6…

A: We have to find the firstd erivative of the given functions.Recall some basic formulas: ddxxn=nxn-1…

Q: 4

A:

Q: Let Var(X₁) = Var(X3) = 2, Var(X₂) = 3, Cov(X₁, X₂) = 2, Cov(X2, X3) -2, Cov(X1, X3) = -1. i)…

A:

Q: (b). Use the definition of the derivative to evaluate f(x)=x". Hint: Use the Binomial Theorem:…

A: b) Given function is f(x)=xn  We have to find the derivative of f(x)=xn by using the definition of…

Q: None

A: Given, a)   f(x)=(x-2)2  b)   f(x)=(x-1)(x+1)

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

A:

Q: Assume that h[n]=(-1)"g[n]. What is the relation between H(2) and G(z) o HZ) = (- 1)" G(z) H(2)…

A:

Q: In (z z2) =

A: I have provided this answer in step-2.

Q: Note that 1/4 ≠ 1/2; but (1/4)^(1/4) = (1/2)^(1/2). Prove that there exists infinitely many pairs of…

A: We will first parametrize the equation : aa = bb  Then we will eliminate the trivial solution : a =…

Q: LET • (CX)= In [x²(x+5) ³ (x² + 4)⁰7 f'(x) = LET F(x) = \n [X²(X + 3) ³ (X² + 2)²] Cico

A:

Q: How do I find the syndrome polynomial s(x) = w(x) mod g(x), where g(x) = 1+x+x^2+x^4+x^8 and w(x) =…

A: To find the syndrome polynomial s(x) given  and, we need to perform polynomial division w(x)÷g(x)…

Q: 1/(1+x*) 8) f(x)={-1/(1+ x') ーTくX<0 Ans. Odd 0<x<π

A:

Q: state formally the Heine-Borel Theorem in R^k

A: Let S be a subset of ℝk. Then the Heine-Borel theorem states :