Question

Using the function f(x)=ln(1+x)

a. Find the 8 degree taylor polynomial centered at 0 and simplify.

b. using your 8th degree taylor polynomial and taylors inequality, find the magnitude of the maximum possible error on [0,0.1]

c.approximate ln(1.1) using your 8th degree taylor polynomial. what is the actual error? is it smaller than your estimated error?Round answer to enough decimal places so you can determine.

d. create a plot of the function f(x)=ln(1+x) along with your taylor polynomial. Based on the plot what appears to be the interval of convergence? explain.

Answer 1

(a) The Taylor series of a function f(x) at x = 0 is called the Maclaurin series of the function, which is given by

Now, our given function is f(x) = ln(1+x) so we have

So we have

Simplifying, we get

Now, we need to take the 8th degree Maclaurin series, so we take upto 8 terms of the above series. Hence we have

which is our required answer.

(b) Now, Taylor's inequality gives us an upper bound of the error of our approximation. We have

where M is the upper bound of the (n+1)-th derivative in the given region, that is

Now, we are considering the 8th degree Taylor polynomial around 0, so in our case a = 0 and n = 8. So we have

..................(1)

and we have

So, in the interval [0,0,1], the highest value of this function will be when the denominator is lowest, which happens when x = 0 so

Thus, expression (1) becomes

Now, in the interval [0,0,1], the right hand side is highest when x = 0.1 so

which is our required answer.

(c) Now, let us find ln(1.1) using our Taylor polynomial. For this, we have to put x = 0.1, so that 1+x = 1.1. Hence we have

which gives

Now, using a calculator, we see that the actual value of ln(1.1) is exactly the same as our approximation upto all the decimal places calculated. Hence it is smaller than our estimated error.

(d) Drawing our Taylor polynomial in red and drawing the function f(x) using dotted blue lines, we see that

Our Taylor polynomial clearly manages to properly give an estimate of f(x) when x is roughly inside the interval -1 < x < 1. Hence this appears to be the interval of convergence of the given function with respect to the obtained Taylor polynomial.