Using induction, prove that `2' = 2"+1 – 1. i=0 Using induction, prove that En n! = (n + 1)! – 1. %3D i=1

Q: W hich of the terms in the expansion of (y³ +y')" will involve y? 6th term 7th term 8th term 9th…

A: Given that:.     (y3 +y-1)10 . First we find expansion of this and then we find term in this involve…

Q: If z1=1-i,z2=-2+4i then evaluate: modulus of (z1+z2+1)/(z1-z2+i)

A:

Q: A function () of period defined by 16)-1²-1 Obtain a Fourier series expansion of the function. -*…

A: Given function is ft=t2−t and −π<t<π. We have to find the Fourier series expansion of the…

Q: ded as a Fourier se of x, then

A:

Q: Find the Fourier series for the etion fon) = |x\, -2{*<! %3D what happens at x==2 and x=0 ?

A: Period  T =3 Frequency  ω=2πT=2π3 a=-2, b=1 The Fourier series is a0  +    ∑K=1∞ ancosKωx+bnsinKωx…

Q: (10) Find Std(X) when X is a discrete random variable with the following given density. (i) f(z) =…

A: Solution Dear student  as  per guidelines we answer the first question only when multiple different…

Q: Determne the |laylur Series of flx) = Sh(x)= X =t. Series of Flo) = Sin(x)= arownd Xo

A:

Q: 10. Let [ A= 103 -1 21 02 Calculate det (A) by cofactor expansion.

A:

Q: Prove that the sum from i=1 to n of 1/(4i2-1)= n/(2n+1) for n=1,2,3 by (b) Partial fractions and…

A:

Q: Prove or disprove the following claim using a proof by induction. Claim: Vn € Z>1, [n even ^ (−1)n =…

A: We have the claim ∀n∈ℤ≥1, n even∧-1n=1∨n odd∧-1n=-1 This can be written as: for all integers greater…

Q: Prove: For any positive integer n, x™1 - yl = (x- y) xy' (You may use the laws | of exponent in your…

A:

Q: (7) Use congruences to prove the following. They were proven by induction in Prob- lem 19 in Section…

A: (a) To prove that 9 divides (4^n+15n-1), we need to show that (4^n+15n-1) is congruent to 0 mod 9.…

Q: Use a Taylor expansion around 'a' to prove that In x = In a+x=1 a)". Discuss all the cases of given…

A: Introduction: The Taylor series expansion is an application of successive differentiation. The…

Q: Determint the function in the interval fourier series f(x)=2x-20 of the in the -3<x<3

A: As per the question we are given a periodic function f(x) = 2x - 20 for, -3≤x≤3 And we have to find…

Q: Prove by induction that, for every n\geq2 n+1 (1-2)(1-¹)(1-(1-1/2) = ² 2n

A: To Prove: We have to prove by induction that for every n≥2, we have:…

Q: If n^2+6n+11 = x^2+y^2 for positive integer n, x and y prove [xy/2] is composite. Note: [t]…

A: Given :              n2+6n+11 = x2+y2              t ≤ t

Q: Let f(x) = 1- x on 0, T|. The Fourier cosine series of f on 0, T is a) None of these (-1)" cos (! b)…

A:

Q: o<×くT ② Foのコ 2 [:

A:

Q: *prove that 1P4 ľ +3'+ ... +n² = n' (n+1)* 4 for n= l,2,3

A: The given expression is the series expression. Formula to find the sum of series helps to prove the…

Q: Consider the function f defined on the interval [-4, 4] as follows, -5, f(x) = { Denote by f the…

A:

Q: Fourier series f(x)= x sinx 2 xe (7)

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: 69. Use the principle of mathematical induction to prove that dk : (In x) = (–1)k-1 «– D! for k > 1.…

A: Use the principal  induction to prove thatdkdxk(ln x)=(-1)k-1(k-1)!xk , for k≥1.

Q: 3: You are given that the Fourier series expansion of f(x) = x2, x € (-7, 7) with period 27 is +4…

A:

Q: What is the complex Fourier series of the function f(t) = cos t? Hint: if you're confident, you can…

A: We find the complex Fourier series of the function f(t)=cos(t).

Q: The complex Fourier series of a function f(x) is given by 1+in einz. Then | (2)e* dz = %3D 1+ in4…

A: The complex Fourier series of a function fx is given by  fx=∑n=-∞∞1+in1+in4einx…

Q: The generating function of a_{n+1}=8a_{n}+4*, where a,=3 is Ο a. Σ2x)^+Σ (4x) ^Κ O b.…

A: Given an+1=8an+4n+1 Multiply the recurrence relation by xn an+1xn=8anxn+4n+1xn…

Q: How did they go from 30(2^k-1) to 15(2^k)? Please explain

A:

Q: . Let n be a positive integer. Prove that 1 · ( n/ 1)  + 2 · ( n/ 2)  + . . . + n ·  (n/ n) …

A:

Q: 21) f(x)=47x A) (4 ln 7)47x Write summation notation for B) (7ln 4)47x x(In 2) C) (In 4) 47x x(In x)…

A: Finding the derivative of f(x)

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

A:

Q: a) y tan-¹x b) y = -x Find the MACLAURIN EXPANSION of the 2 functions above. (Use at least 5 terms)

A: Maclaurin series of f is given by, ∑k=0∞fk(0)k!xk =f(0) +f'(0)x + f''(0)2!x2 + . . . + fn(0)n!xn + .…

Q: 13) Expand fit) in Fourier series in the interve) (-2, 2) when fit) = -zst <o

A:

Q: 9- a) b) C) e) The Fourier series of the function f(x) + 2 8 0 -4 = 8, -4, -π<x<0 0<x<T at the point…

A:

Q: 6) E(-1y " Ans. Diverges, a,1 n+1

A:

Q: Proof using induction method: a+ar+ar+ar'+... . +ar"=(ar" 1-a)/(r-1)

A: To prove the formula:

Q: The Taylor expansion of f for x→ 10 is f(x) = 10 + 3(x-10)² + 2(x-10)³ + 0((x a) ft (10) = -12 6) fr…

A:

Q: expanded as odulus of x, the 1-e

A: