By using several classic examples,this paper discusses the application of Fourier series in the summation of Multinomial se-ries,ordinary differential equations and wave equations.By choosing a reasonable function and expanding it into a Fourier series,the val-ue of the Fourier series at a particular point is found as a sum of several terms of the series.The general solution of a second-order ordi-nary differential equation is considered as the form of a Fourier series,by the method of coefficients to be determined,it is derived.For a wave equation with an initial margin problem,a nontrivial special solution with a Fourier series is derived by the variable substitution method,and the Fourier coefficients are derived by utilizing a term-by-term method of derivatives and integrals to arrive at a special so-lution of this wave equation.
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