# The function is g(x)= 2x lnx -1 Use Newton-Raphson formula to show this function can be expressed as xn+1 = (xn + 1/2)/(1+lnxn) Thanks

rcmath | High School Teacher | (Level 1) Associate Educator

Posted on

Newton-Raphson formula states that

`x_(n+1)=x_n-[g(x_n)]/[g'(x_n)]`

Let's find g'(x)

`g'(x)=2lnx+2x*1/x=2lnx+2`

Now let's plug everything in the formula

`x_(n+1)=x_n-[2x_nlnx_n-1]/[2lnx_n+2]`

Dividing the numerator and denominator by 2 give us

`x_(n+1)=x_n-[x_nlnx_n-1/2]/[lnx_n+1]=>`

`x_(n+1)=[x_n(lnx_n+1)-x_nlnx_n+1/2]/[lnx_n+1] =>`

`x_(n+1)=[x_nlnx_n+x_n-x_nlnx_n+1/2]/[lnx_n+1]=>`

`x_(n+1)=[x_n+1/2]/[lnx_n+1]`