I've been struggling with finding Fourier series of the given function for a while now
I've calculated my coefficients using formulas :
Though my results does approximate function sufficiently enough (see the picture), I'm pretty sure that I did something wrong, as online calculators give me different values for coefficients, and the sum of some series(like (-1)^n/n^2) does not match expected values(integrals are calculated correctly). Is there something wrong with how I approach solving those integrals for coefficients or maybe there are some other pitfalls that I could have encountered? I would really appreciate it if you could help me out at least with calculating . Thank you in advance.
Resulting Furier series
Answer
Here are my calculations. I haven't thoroughly checked them, but hopefully they will help.
an=12[∫1−2(1−x)cosnπ2x−∫21(1−x)cosnπ2x]12[2(x−1)nπsinnπ2x+4n2π2cosnπ2x|21+2(x−1)nπsinnπ2x+4n2π2cosnπ2x|−21]
Evaluated at x=2, and x=−2,
sinnπ2x=0cosnπ2x=(−1)n
at x=1,(x−1)=0
cosnπ2x=0 if n is odd, −1 if n≡2(mod4), 1 if n≡4(mod4)
an=−4n2π2,8n2π2,−4n2π2,0 when n≡1,2,3,4 respectively.
bn=12[∫1−2(1−x)sinnπ2x−∫21(1−x)sinnπ2x]12[4n2π2sinnπ2x−2(x−1)nπcosnπ2x]
Evaluated at:
x=22nπcosnπ(−1)n2nπ
x=−26nπcosnπ(−1)n6nπ
x=14n2π2sinnπ24n2π2,0,−4n2π2,0
bn=−4nπ−4n2π,4nπ,−4nπ+4n2π,4nπ
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