PYQs Based Test - 07 : Taylor and Fourier Series

The Fourier series of the function,

f(x) = 0, -π < x ≤ 0

= π - x, 0 < x < π

in the interval [= π, π] is

$f\left(x\right)=\frac{\pi }{4}+\frac{2}{\pi }\left[\frac{\cos x}{1^2}+\frac{\cos 3x}{3^2}\dots \dots \dots \right]+\left[\frac{\sin x}{1}+\frac{\sin 2x}{2}+\frac{\sin 3x}{3}+\dots \dots \dots \right]$

The convergence of the above Fourier series at 𝑥 = 0 gives

The quadratic approximation of (x) = x³- 3x²- 5 at the point x = 0 is

The Fourier cosine series for an even function f(x) is given by

$f(x)=a_0+\sum _{n=1}^{\infty }{a}_n\cos (nx)$

The value of the coefficient a₂ for the function f(x) = cos²(x) in [0,π] is

Taylor series expansion of $\mathrm{f}\left(\mathrm{x}\right)=\underset{0}{\overset{\mathrm{x}}{\int }}{\mathrm{e}}^{-\left(\frac{{\mathrm{t}}^2}{2}\right)}\mathrm{dt}$ around x = 0 has the form

f(x) = a₀ + a₁x + a₂x² + ….

The coefficient a₂ (correct to two decimal places) is equal to ________

In the Taylor’s series expansion of e${}^x$ about x = 2, the coefficient of (x – 2)⁴ is