The Taylor series expansion of sin x-x- at x- is given by

f(x) $= \frac{π}{4} + \frac{2}{π} [\frac{cos x}{1^{2}} + \frac{cos 3 x}{3^{2}} + . . . .] + [\frac{sin x}{1} + \frac{sin 2 x}{2} + \frac{sin 3 x}{3} + . . . .]$ The convergence of the above Fourier series at x = 0 gives

Let $f (x) = {\begin{matrix} - π, if & - π < x \leq 0 π, if & 0 < x \leq π \end{matrix}$
be a periodic function of period $2 π$ . The coefficient of $sin 5 x$ in the Fourier series expansion of f(x) in the interval $[- π, π]$ is

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The Taylor series expansion of sinxx−π at x=π is given by

The correct option is D −1+(x−π)23!+...sinxx−π=sin(π+x−π)x−π =−sin(x−π)(x−π) =−1(x−π)[(x−π)−(x−π)33!+...] =−1+(x−π)23!.....

The Taylor series expansion of $\frac{s i n x}{x - π}$ at $x = π$ is given by

The correct option is D $- 1 + \frac{(x - π)^{2}}{3!} + . . .$
$\frac{s i n x}{x - π} = \frac{s i n (π + x - π)}{x - π}$
$= \frac{- s i n (x - π)}{(x - π)}$
$= \frac{- 1}{(x - π)} [(x - π) - \frac{(x - π)^{3}}{3!} + . . .]$
$= - 1 + \frac{(x - π)^{2}}{3!} . . . . .$