證明cos x=1-(x^2)/2!+(x^4)/4!-(x^6)/6!+(x^8)/8!-(x^10)/10!+...?最好有解釋。?

Sol
fn(x) 代表f(x) n次微分
泰勒展開式
f(x)=f(0)+f1(0)*x+f2(0)x^2/2!+f3(0)x^3/3!+f4(0)x^4/4!+f5(0)x^5/5!
+f6(0)x^6/6!+f7(0)x^7/7!+f8(x)x^8/8!+…….
當f(x)=Cosx
f1(x)=－Sinx
f2(x)=－Cosx
f3(x)=Sinx
f4(x)=Cosx
f5(x)=－Sinx
f6(x)=－Cosx
f7(x)=Sinx
f8(x)=Cosx
…
f(0)=1
f1(0)=0
f2(x)=－1
f3(x)=0
f4(x)=1
f5(x)=0
f6(x)=－1
f7(x)=0
f8(x)=1
…
So
Cosx=1+0x－x^2/2!+0x^3/3!+x^4/4!+0x^5/5!－x^6/6!+0x^7/7!+x^8/8!+…….
=1－x^2/2!+x^4/4!－x^6/6!+x^8/8!+….