Taylor series is an example of infinite series derived from the expansion of `f(x)` about a single point. It is represented by infinite sum of `f^n(x)` centered at `x=c` . The general formula for Taylor series is:
`f(x) = sum_(n=0)^oo (f^n(c))/(n!) (x-c)^n`
or
`f(x) =f(c)+f'(c)(x-c) +(f^2(c))/(2!)(x-c)^2 +(f^3(c))/(3!)(x-c)^3 +(f^4(c))/(4!)(x-c)^4 +...`
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apply the definition of Taylor series for the given function `f(x) = cos(x)` centered at `x=pi/4` , we list `f^n(x)` using the derivative formula for trigonometric functions: `d/(dx) sin(x) = cos(x)` and `d/(dx) cos(x)= -sin(x)` .
`f(x) =cos(x)`
`f'(x) = d/(dx) cos(x)`
`= -sin(x)`
`f^2(x) = d/(dx) -sin(x)`
`=-1*d/(dx) sin(x)`
`= -1 * cos(x)`
`= -cos(x)`
`f^3(x) = d/(dx) -cos(x)`
`=-1*d/(dx) cos(x)`
`= -1 * (-sin(x))`
`= sin(x)`
`f^4(x) = d/(dx) sin(x)`
`= cos(x)`
Plug-in `x=pi/4` on each `f^n(x)` , we get:
`f(pi/4)= cos(pi/4) =sqrt(2)/2`
`f'(pi/4)= -sin(pi/4)=-sqrt(2)/2`
`f^2(pi/4)= -cos(pi/4)=-sqrt(2)/2`
`f^3(pi/4)= sin(pi/4)=sqrt(2)/2`
`f^4(pi/4) =sin(pi/4) =sqrt(2)/2`
Note: `sin(pi/4) =sqrt(2)/2` and `cos(pi/4)=sqrt(2)/2` .
Plug-in the values on the formula for Taylor series, we get:
`sin(x) =sum_(n=0)^oo (f^n(pi/4))/(n!) (x-pi/4)^n`
` =f(pi/4)+f'(pi/4)(x-pi/4) +(f^2(pi/4))/(2!)(x-pi/4)^2 +(f^3(pi/4))/(3!)(x-pi/4)^3 +(f^4(pi/4))/(4!)(x-pi/4)^4 +...`
` = sqrt(2)/2+(-sqrt(2)/2)*(x-pi/4) +(-sqrt(2)/2)/(2!)(x-pi/4)^2 +(sqrt(2)/2)/(3!)(x-pi/4)^3 +(sqrt(2)/2)/(4!)(x-pi/4)^4 +...`
` = sqrt(2)/2-sqrt(2)/2(x-pi/4)-(sqrt(2)/2)/2(x-pi/4)^2+(sqrt(2)/2)/6(x-pi/4)^3 +(sqrt(2)/2)/24(x-pi/4)^4 +...`
` = sqrt(2)/2-sqrt(2)/2(x-pi/4)-sqrt(2)/4(x-pi/4)^2+sqrt(2)/12(x-pi/4)^3 + sqrt(2)/48(x-pi/4)^4 +...`
The Taylor series for the given function `f(x)=cos(x)` centered at `c=pi/4` will be:
`cos(x)= sqrt(2)/2-sqrt(2)/2(x-pi/4)-sqrt(2)/4(x-pi/4)^2+sqrt(2)/12(x-pi/4)^3 + sqrt(2)/48(x-pi/4)^4 +...`