Find the half-range cosine series for the function f(x) = x(n - r) in o<r<n. Deduce...

Find the half-range cosine series for the function f(x) = x(n - r) in o<r<n. Deduce that 1/1⁴+1/2⁴+1/3^4+…. =π⁴/90.

Expert Solution

milcah answered 2 years ago

Find the half-range expansions of the given function. To illustrate the convergence of the cosine and...

Find the half-range expansions of the given function. To illustrate the convergence of the cosine and sine series, plot several partial sums of each and comment on the graph (using words). f(x) = 1 if 0 < x < 1. f(x) = π-x if 0 ≤ x ≤ π. f(x) = x2 if 0 < x <1.

Find the Maclaurin series of the function f(x)=(3x2)e^−4x (f(x)=∑n=0∞cnxn)

Expand in Fourier series: Expand in fourier sine and fourier cosine series of: f(x) = x(L-x),...

Expand in Fourier series: Expand in fourier sine and fourier cosine series of: f(x) = x(L-x), 0<x<L Expand in fourier cosine series: f(x) = sinx, 0<x<pi Expand in fourier series f(x) = 2pi*x-x^2, 0<x<2pi, assuming that f is periodic of period 2pi, that is, f(x+2pi)=f(x)

Find the Fourier Series for the function defined over -5 < x < 5 f(x) =...

Find the Fourier Series for the function defined over -5 < x < 5 f(x) = -2 when -5<x<0 and f(x) = 3 when 0<x<5 You can use either the real or complex form but must show work. Plot on Desmos the first 10 terms of the series along with the original function.

6. (a) let f : R → R be a function defined by f(x) = x...

6. (a) let f : R → R be a function defined by f(x) = x + 4 if x ≤ 1 ax + b if 1 < x ≤ 3 3x x 8 if x > 3 Find the values of a and b that makes f(x) continuous on R. [10 marks] (b) Find the derivative of f(x) = tann 1 1 ∞x 1 + x . [15 marks] (c) Find f 0 (x) using logarithmic differentiation, where f(x)...

Suppose that f(x)=x^n+a_(n-1) x^(n-1)+⋯+a_0∈Z[x]. If r is rational and x-r divides f(x), prove that r is...

Suppose that f(x)=x^n+a_(n-1) x^(n-1)+⋯+a_0∈Z[x]. If r is rational and x-r divides f(x), prove that r is an integer.

What is the Maclaurin series for the function f(x)=arcsinx. Find the radius of convergence of the...

What is the Maclaurin series for the function f(x)=arcsinx. Find the radius of convergence of the series.

Find the Fourier series for the function on the interval (-π,π) f(x) = 1 -sin(x)+3cos(2x) f(x...

Find the Fourier series for the function on the interval (-π,π) f(x) = 1 -sin(x)+3cos(2x) f(x )= cos2(x) f(x) = sin2(x)

A probability density function on R is a function f :R -> R satisfying (i) f(x)≥0...

A probability density function on R is a function f :R -> R satisfying (i) f(x)≥0 or all x e R and (ii) \int_(-\infty )^(\infty ) f(x)dx = 1. For which value(s) of k e R is the function f(x)= e^(-x^(2))\root(3)(k^(5)) a probability density function? Explain.

Find the Fourier cosine and sine series of f(t) = t^2 ,0 < t < π

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