T=4
a0=41∫−13f(x)dx=41∫03dx
=41[x]30=43
an=41∫−13f(x)cos(2nπx)dt=41∫03cos(2nπx)dx
=2nπ1[sin(2nπx)]30=2nπ1sin(23nπ)
n=1,a1=−2(1)π1
n=2,a2=0
n=3,a3=2(3)π1
...
a2k+1=2(2k+1)π(−1)k+1,k=0,1,2,...
bn=41∫−13f(x)sin(2nπx)dx=41∫03sin(2nπx)dx
=−2nπ1[cos(2nπx)]30=2nπ1(1−cos(23nπ))
n=1,b1=2(1)π1
n=2,b2=2(2)π2
n=3,b3=2(3)π1
n=4,b4=0
n=5,b5=2(5)π1
b6=2(6)π2
...
b2k+1=2(2k+1)π1,k=0,1,2,...
b4m+2=2(4m+2)π2,m=0,1,2,..
f(x)=83+k=0∑∞2(2k+1)π(−1)k+1cos(2(2k+1)πx)
+k=0∑∞2(2k+1)π1sin(2(2k+1)πx)
+m=0∑∞2(2m+1)π1sin((2m+1)πx)
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