The goal of this problem is to approximate the value of ln 2. We will use two different approaches: (a) First, we use the Taylor polynomial pn(x) of the function f(x) = lnx centered at a = 1.
- Write the general expression for the nth Taylor polynomial pn(x) for f(x) = lnx centered at a = 1.
- At x = 2, evaluate the size of the remainder Rn(2) = ln 2 − pn(2).
- What should n be so that you are sure that pn(2) approximates ln2 to two decimal
points? What is then the approximate value of ln 2 (up to two decimal points)?
The Attempt at a Solution
For part 1, I netted the (should be correct) answer of
ln(x) = Σ from n=1 to infinity of (-1)^(n+1)/n (x-1)^n
Now, I am completely stuck on part 2. At x=2, evaluate the size of the remainder Rn(2) = ln(2)-Pn(2).
Are there any examples out there? I am searching the internet for examples, but not much luck.