Q: If C,,C,, C,., C, denote the binomial coefficients in the expansion of (1+x)", prove that (-1) 1+r log,10 = 0 (1+ log, 10*)' r=0

Q: Find the Taylor series expansion of f(x) = ln(3-2x) about a = 1.

A: Given: f(x)=ln3-2x,a=1

Q: 2. Find the exact value of the term containing x" in the expansion of (x-3y)" using the Binomial…

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Q: nsion of φ(x) = 1/(1−x)

A: The given expression is φ=1(1-x), when n=4 and x0=-1

Q: Determine as, az, and as What is the value of b. Express q(t) as a Fourier series expansion

A: The given function is q(t)=t               for 0≤t≤π2π-t       for π≤t≤2π From the graph, it can be…

Q: (1 point) Write the Taylor series for f(x) = sin(x) at z as %3D Find the first five coefficients.…

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Q: Find the Fourier Series expansion of (-) - 3 < x <0 f(x) = %3D 0<x < 3 And f(x) = f(x+6).

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Q: ] Find the Taylor series expansion of f(x) = ln(3-2x) about a = 1.

A: The general formula for the derivatives is  dxndx=nxn-1 A taylor series is given by…

Q: 2. Use the Binomial Theorem for the expansion of the function as a power series f(x) = {1+3x²

A: Given  f(x)=1+3x23

Q: 3 The Taylor series of f(x)=is generated at x=2.

A:

Q: Find the Fourier series expansion for: x2 f(x) = x + ,-n <x < n 4

A: Let us consider the function f(x) on the interval [-L,L] is obtained as…

Q: Show that the Taylor series for ƒ(x) = tan-1 x diverges for  |x|&gt;1.

A: Using Taylor Series.

Q: 18. Prove that the zeta-function series S(x) - k=1

A: To prove that the zeta-function series                                 ζ(x):-∑k=1∞1kxconverges for…

Q: Expand in a Fourier series f (x) = a², 1 < x < 1 then find 2' n4

A: Given that fx=x2 and find Fourier series in -12&lt;x&lt;12. The Fourier series is…

Q: c) Use Taylor series expansion of f(x) = In(1 + cos x) given by 1 f(x) = In 2-x²- *** 1440 to…

A: c Given that the Taylor series expansion of the function fx=ln1+cosx is given by…

Q: 2. Use the Binomial Theorem for the expansion of the function as a power series. _f (x)= {/1+3x²

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Q: 2) Obtain the Fourier series expansion of the below function, f(x) = {2(k – x), 0, 0 <x<n %3D n<x<…

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Q: Find the Taylor series expansion of f(x)= 1/(9x+4)2 at x=-2 for n=0 to n=6

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Q: What is the Taylor's expansion of (3/x^4) about x=1 up to the first five non-zero terms?

A:

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: Is there any alternative way to find the expansion of pi other than using the Taylor Series model?…

A: Yes, there is an alternative way to find the expansion of pi other than using the Taylor Series…

Q: he first four terms in the Maclaurin series expansion for f(x) = ln( (1+2x) / (1+3x) )  are

A:

Q: 9. Given the Mac expansion of y = In(1+ x) below, use it to approximate In(0.4) to the 4th decimal…

A: To approximate ln(0.4) to the 4th decimal place accuracy using the Maclaurin expansion of y=ln1+x.

Q: Find the coefficient of x* in the power series 4 3x² +2x expansion of x² +2)(x-3)

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Q: [2, if - 2<z < 0 1, if 0 <r< 2 3. Find the fourier series expansion for f(r) = and show that 1 ( a)…

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Q: 5. Expand the following function in Fourier series where f(x)=x+r and show that 1 =1-- + 3 1 1 + 1 5…

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Q: 5. Find the Taylor expansion of f(x)=x² - 2x + 4 at x = 2.

A: To find the Taylor expansion of fx=x2-2x+4 at x=2.

Q: Whar is the Taylor series representation for f(x)=ln(1+4x)?

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Q: Find the coefficient of x° in the power series 3. expansion of 2 x+2

A: Use expansion of (1+x)^-n to find the coefficient of x^6 .

Q: Q 2. If f (x) = x in (-5) and f (x) = O in (,") find the Fourier series of f(x). Deduce that 1 Ln=1…

A: We know that the Fourier series of function f(x) is given by where

Q: (2, if -2x <0 f (x)=- x, if 0<x<2 %3D

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Q: Find a Taylor series expansion around x = 2 for f(x) = ln x

A: To find the Taylor series expansion for f(x)=ln(x) around x=2 is given by :-

Q: Give a recursive definition (with initial condition(s)) of (an) where (n = 1, 2, 3, . . . )

A: We have, an = 212nWe have to find a recursive definition with initial conditions of an starting from…

Q: Find the Fourier Series Expansion of the following functions: if – 2<x <0 Sx + 2, f(x) = lx- 2, if…

A: Since you have asked multiple questions so as per guidelines we will solve the first question for…

Q: 2. Obtain the Taylor's series expansion of log(cosx) about x = 7% upto the fourth degree.

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Q: Write the Taylor series for f(x) = x³ at x = -4 as > en (x + 4)". n=0

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Q: 6. (a) Express the function f(x) = x in on [0, T] as a Fourier cosine series S(x). What is S(-1)?

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Q: 5 If f(x) is gen then use Fourier series to prove that b, =0

A: We have f(x) is even .

Q: 3) Find the following in the Fourier series expansion of the following functions: a. ao for f(x) =…

A: The given question is solved below.

Q: (* if -<x<! (o if 플<x< Find the Fourier series expansion of f(x) = 2

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Q: 2. Use Taylor series expansions to derive the error term of the sided difference ap- proximation to…

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Q: 2. Use the Binomial Theorem for the expansion of the function as a power ser: S(x) = V1+ 3x²

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Q: 4. Does the Taylor series expansion of f(x) = (1 + 3x)24 about a = 1 have only a finite number of…

A: Solution

Q: Find the binomial expansion of √(x² +1-y²) up to the fifth term.

A: As per the question we are given a bivariate function in variables x, y and we have to find its…

Q: Using binomial expansion, prove that z3 = [r (cos 0 + isin 0)]3 is equal to r3 (cos 30 + isin 30).

A:

Q: Find the sum of 1+1/1^2 +1/3^2 +1/5^2 +⋯, using Fourier series expansion if f(x) = a when [0,π] and…

A: Given:- The sum 1+112+132+152+............ if f(x)= a when 0,π and 2π-x when π,2π

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

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Q: 3. Find the Fourier series of, - n<x<-1/2 f(x) = }0 - 1/2 <x <n/2 1/2 < x <n 1

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Q: 6) Use the first three terms of the Taylor expansion for f(x) = cos(x) to estimate cos(1/8). %3D

A: We have to find the value of cos(1/8)using only first three terms of Taylor's series.