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Question Number 26023 by abdo imad last updated on 17/Dec/17

$$answertoquestion25980keyofslutionwedevelopthe$$$$foctionf\left(x\right)=sin\left(px\right)atfourierserie(\left(f2\pi periodic\right)$$$$f\left(x\right)=\sum _{n=1}^{n=\propto }{a}_{n}sin\left(nx\right)and{a}_n=2/T{\int }_{\left[T\right]}sin\left(px\right)sin\left(nx\right)dx(T=2\pi )$$$$$$$$$$$$$$$$a_n=2{\pi }^{-1}{\int }_0^{\pi }sin\left(px\right)sin\left(nx\right)dx->a_n={(-1)}^{n}sin\left(p\pi \right).2n{\pi }^{-1}{(n^2-p^2)}^{-1}$$$$-->sin\left(px\right)=2sin\left(p\pi \right).{\pi }^{-1}\sum _{n=1}^{\propto }n{(-1)}^{n-1}{(n^2-p^2)}^{-1}sin\left(nx\right)$$$$=2{\pi }^{-1}sin\left(p\pi \right)({(1^2-p^2)}^{-1}sin\left(x\right)-2{(2^2-p^2)}^{-1}sin\left(2x\right)+\dots )$$

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