`f(x)=1/x ,c=1` Use the definition of Taylor series to find the Taylor series, centered at c for the function.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

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 +...`

To apply the definition of Taylor series for the given function `f(x) = 1/x` centered at `c=1` , we list `f^n(x)` using the  Power rule for differentiation: `d/(dx) x^n= n *x^(n-1)` and basic differentiation property: `d/(dx) c* f(x)= c * d/(dx) f(x)` .

`f(x) =1/x`

`f'(x) = d/(dx) 1/x`

       `= d/(dx) x^(-1)`

       `=-1 *x^(-1-1)`

       `=-x^(-2) or -1/x^2`

`f^2(x)= d/(dx) -x^(-2)`

            `=-1 *d/(dx) x^(-2)`

            `=-1 *(-2x^(-2-1))`

           `=2x^(-3) or 2/x^3`

`f^3(x)= d/(dx) 2x^(-3)`

           `=2 *d/(dx) x^(-3)`

          `=2 *(-3x^(-3-1))`

          `=-6x^(-4) or -6/x^4`

`f^4(x)= d/(dx) -6x^(-4)`

            `=-6 *d/(dx) x^(-4)`

            `=-6 *(-4x^(-4-1))`

           `=24x^(-5) or 24/x^5`

Plug-in `x=1` , we get:

`f(1)=1/1 =1`

`f'(1)=-1/1^2 = -1`

`f^2(1)=2/1^3 =2`

`f^3(1)=-6/1^4 = -6`

`f^4(1)=24/1^5 = 24`

Plug-in the values on the formula for Taylor series, we get:

`1/x =sum_(n=0)^oo (f^n(1))/(n!) (x-1)^n`

` =f(1)+f'(1)(x-1) +(f^2(1))/(2!)(x-1)^2 +(f^3(1))/(3!)(x-1)^3 +(f^4(1))/(4!)(x-1)^4 +...`

` =1+(-1)*(x-1) +2/(2!)(x-1)^2 +(-6)/(3!)(x-1)^3 +24/(4!)(x-1)^4 +...`

` =1-(x-1) +2/2(x-1)^2 -6/6(x-1)^3 +24/24(x-1)^4 +...`

` = 1-(x-1)+ (x-1)^2 -(x-1)^3 + (x-1)^4 +...`

The Taylor series for the given function `f(x)=1/x` centered at `c=1` will be:

`1/x=1-(x-1)+ (x-1)^2 -(x-1)^3 + (x-1)^4 +...`

or

`1/x =sum_(n=0)^oo(-1)^n(x-1)^n`

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial