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

Q: 1 The 2nd degree Taylor polynomial for the function f(x) = In x, xo = 3 is P2(x) = In(3) + ÷(x– 3) +…

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Q: n(n + 1)(2n + 1) 2. Use induction to prove that )k2 for all n e Z+. %3D 6. k=1

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Q: 25: Using a suitable Fourier- Series expansion for 0 S x ST: A) Show that: x(n -x) (cos 2x cos 4x .…

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Q: 2: Find the Fourier series of the function for n >t >0 for 2n > t > n 2 f(t) = { , } -2

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Q: a. Use sigma notation to write the Taylor series T(x) about xo = * for the function: f(x) = cos(7x).…

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Q: 7) Find the Fourier series of the following a) f(x) = , |x| < n 4

A: We have to find fourier series of given function-

Q: Applying the Cauchy's Residue Theorem, evaluate de (5 – 3 sin 0)²

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Q: 2. (a) Use Taylor's theorem to find the first degree expansions of (i) sin x, (ii) e and (iii)…

A: for the function  f(x)  taylor polynomail of first degree about  x=0 is p(x)=f(0)+x f'(0)

Q: Prove that for all positive integers n.

A: To prove for all positive integer n ∫01(lnx)ndx=(-1)nn!

Q: (a) Assume that f(x) is function defined by -4 if -T≤ x < 0, f(x) = for 0≤x≤T. (i) Expand the…

A: According to guidelines we can solve only one problem

Q: 6. Find the nth Taylor polynomial centered at C a. f(x) = x²cos x, n = 2, c = n

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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: Q2: Find the Fourier series of the function. if – n<t < 0 2 r(t) = r(t + 2n) = r(t) if 0<t <n 2 --

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Q: 7) Find the Fourier series of the following: a) f(x) %3 х, х E [0, 2п]

A: The Fourier series of f(x) over [0,2π] is

Q: A. Find the Fourier series expansion for: f(x) = x +- -n SxS T 4

A: We will expand as shown below

Q: ind the Fourier series representation of the even extension of the function 0025.png 0 < x<1 if…

A: For even extension Fourier series is called as Fourier cosine series and given by    feven(x) =…

Q: The function 5x+1 f(x) 1-In (x³ +e) has a Maclaurin Expansion O false O true

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Q: (a) Find the lower bound of sum E-1 i° using an integral 3D1 (b) manually compute E-1i° for n = 8,…

A: a The sum of a series can be bounded by two integrals as shown: Let 'S' be the sum of series…

Q: (12) The coefficients (a, ) of the Fourier series of the function f(x)= x;-7<x<a, is

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Q: 1 What is the expansion of f (x) = 1-r OE(-1)"x" Ο Σ 2n

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Q: Prove that for all x > 0 and all positive integers n x2 e* > 1+x + 2! +3 +: 3! x" + n! Recall that…

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Q: 6. What is the coefficient of x'y" in the expansion (4x + 5y) s. Show your work here:

A: We find coefficient of x7y8 in the expansion (4x+5y)15

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: (b) x[n] is a periodic signal with No = 8 and the Fourier series coefficients 0, X [k] = k (mod 4) =…

A: b) x[n] is periodic with period N0=8 So its Fourier series coefficient is also periodic with period…

Q: a) Find a Fourier series to represent f(x)=x- x² from x=-7 to x=T , and deduce that 1 + 12 12 22 '…

A: (a) Given,                f(x)=x-x2      from x=-π to x=π.

Q: Express f(x) = cos wx, - <x<n, as a Fourier series. Hence prove that 2w 2w coton = 2w + 12)(a)²-22)…

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Q: (iii) Prove that there is a constant C such that 0(2º) 1.

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Q: 5. Find the n-th Taylor polynomials for f(x) = 1 about x = 3 and express x + 2 it in sigma notation.

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Q: Let dg denote the number of derangements of order k. (a) Prove the identity C) d = n! with a…

A: Given dk is number of derangement of order k. We know that for a set 1,2,…,k, the derangement dk are…

Q: (a) Find the Fourier series of the function (x) = 1 – |x|, -2 < x< 2 (b) Hence prove that 1+

A: Given : function         fx=1-x  ,  -2&lt;x≤2 (a) To find : Fourier series of the function above (b)…

Q: If Rolle;s theorem applies on f(x) = 9 – (x – 3)², there exists a 'c' in [0, 6], Which satisfies the…

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Q: 5. (a) Find the Fourier Series of the function (x) = 0 (x) = nx- x if 0< x< n. if -n < x < 0 (b) Use…

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Q: 12. Find the general solution using Reduction of Order: a) y" = In x

A: you have asked multiple questions in a single request we would be answering only the first question…

Q: Bb Announcements - Spring 2020-C X W a-concrete-introduction-to-highe X b My Questions | bartleby…

A: Let

Q: 2. Find the Fourier series of f(x) = x² (-1<x<1), p=2.

A: Fourier series expansion is the method of representing a periodic function as an infinite sum of…

Q: 6. Let f be defined as 0, -1 < x < 0 f(x) = { x +1, 0<x<1 Find the Fourier Series of f (x) on the…

A: Given, fx=0, -1&lt;x&lt;0x+1, 0≤x&lt;1

Q: 1. (a) Find the Fourier series of the function f(x) = 1 |x|, -2<x≤2 (b) Hence prove that 1 + 3/2 +…

A: 1. (a) Given,                 f(x)=1-x, -2&lt;x≤2

Q: 3. Prove that A=I-N+N²_N³ for A & N defined as follows: N=A–I (identity), where 1 0 0 & I=|0 1 0 1…

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Q: 2. Find the Fourier series for the function f(x)= (n – x)² in -n <x < A.

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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…

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Q: 5) If f(x)= x ? ; (x +4)=f (x) b, The coefficient n in this Fourier series is : (-1)". (nz) (-1)** -…

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Q: Q6 Using the Fourier series of f(x) 1 + 1 -+ 2 1 1 1 + = 9 + 1 16 (-π< x < π) Prove that 1 π² 6 + 25…

A: The given function is f(x)=x22 ,   -π&lt;x&lt;π    The Fourier series of a function is of the form…

Q: For f(x) = 1 –x2on [−2, 1], do the hypotheses and conclusion of Rolle’s Theorem hold?

A: Rolle's Theorem: Rolle's theorem states that if a function f is continuous and differentiable on the…

Q: 1. Find the Fourier Series coefficients representation of if -T < x < 0 f(x) TT – x if 0 < x< T…

A: Note:-  Fourier coefficients:- a0 =1L∫-LL fx dxa0 =1L∫-LL fx cosnπxL dxa0 =1L∫-LL fx sinnπxL dx…

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: 5. (a) Find the Fourier Series of the function if -n < x < 0 (x) = 0 (x) = nx- x if 0 < x < n. %3D…

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Q: Let n E Z. Show that f(x) = x3 – nx +2 is irreducible over Q for %3D n + -1,3, 5.

A: We shall find all n ∈ Z such that f(x) is reducible. Assume that f(x) is reducible. As f(x) is…

Q: Q./ Prove( in detail) the solution of (bn) in fourier series is ( -4n / n)

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Q: 5 Is the function f(#) = E (-1)" X[n,n+1) n21 integrable on R in the Lebesgue sense. Here XA is the…

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Q: -- a) such that ab =1. Let p>1 be a prime number. Show that for each a + 0 in Zn there is b in Zp…

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Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products. 

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.