document.write( "Question 30437: Find the Fourier expansion on [pi, pi] of the function f(x)= x \n" ); document.write( "

Algebra.Com's Answer #17185 by venugopalramana(3286) $\"\"$ $\"About$

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Find the Fourier expansion on [pi, pi] of the function f(x)= x
\n" ); document.write( "I THINK YOU MEAN -PI TO PI
\n" ); document.write( "F(X)=X...THIS AN ODD FUNCTION..SO
\n" ); document.write( "F(X)= SIGMA[BN*SIN(NX)]
\n" ); document.write( "BN=(1/PI)*INTEGRAL[-PI TO PI {F(X)SIN(NX)DX}]
\n" ); document.write( "=(1/PI)*INTEGRAL[-PI TO PI {X*SIN(NX)DX}]
\n" ); document.write( "=(1/PI)
\n" ); document.write( "=(1/PI)
\n" ); document.write( "=(1/PI)*{(-PI*COS(N*PI)/N)-(PI*COS(-N*PI)/N + SIN(N*PI)/N^2 - SIN(-N*PI)/N }
\n" ); document.write( "=(1/PI)*(-2PI*COS(N*PI)/N)..SINCE SIN(N*PI)=0..FOR ALL N AND
\n" ); document.write( " COS(-N*PI)=COS(N*PI)
\n" ); document.write( "BN=(1/PI)*{-2PI*(-1)^N}/N...SINCE COS(N*PI)=-1 WHEN N IS ODD AND COS(N*PI)=1
\n" ); document.write( "WHEN N IS EVEN.
\n" ); document.write( "BN=-2(-1)^N /N...HENCE THE SERIES IS
\n" ); document.write( "X= 2{SIN(X)/1 - SIN(2X)/2 + SIN(3X)/3 - SIN(4X)/4 +............} \n" ); document.write( "

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