This problem asks for Taylor polynomials forf(x) = ln(1 +x) centered at= 0. Show Your work in an organized way.(a) Find the 4th, 5th, and 6th degree Taylor polynomials forf(x) = ln(1 +x) centeredata= 0.(b) Find the nth degree Taylor polynomial forf(x) centered at= 0,written in expanded form.

a) The 4th degree , 5th degree and sixth degree polynomials

b)The nth degree Taylor polynomial for f(x) centered at x = 0, in expanded form.

Step-by-step explanation:

Given the polynomial function f(x) = log(1+x) …...(1) centered at x=0

      f(x) = log(1+x) ……(1)

using formula

Differentiating Equation(1) with respective to 'x' we get

 ….(2)

At x= 0

using formula

Again Differentiating Equation(2) with respective to 'x' we get

   ….(3)

At x=0

Again Differentiating Equation(3) with respective to 'x' we get

….(4)

At x=0

Again Differentiating Equation(4) with respective to 'x' we get

....(5)

Again Differentiating Equation(5) with respective to 'x' we get

.....(6)

At x=0

Again Differentiating Equation(6) with respective to 'x' we get

and so on...

The nth term is

Step :-2

Taylors theorem expansion of f(x) is

At x=a =0

Substitute  all values , we get

On simplification we get

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step-by-step explanation: