Question

Assume that f(x)=ln(1 x) is the given function and that pn represents the nth taylor polynomial centered at x=0 . Find the least integer n for which pn(0.6) approximates ln(1.6) to within 0.0001.

Answer 1

To find the least integer n for which pn(0.6) approximates ln(1.6) to within 0.0001, we can use Taylor series.

Step-by-step explanation:

To find the least integer n for which pn(0.6) approximates ln(1.6) to within 0.0001, we can use Taylor series. The nth Taylor polynomial of the function f(x) centered at x=0 can be written as:

pn(x) = f(0) + f'(0)x + (f''(0)x^2)/2! + ... + (f^n(0)x^n)/n!

In this case, f(x) = ln(1 + x), so the nth Taylor polynomial is:

pn(x) = x - (x^2)/2 + (x^3)/3 - ... + ((-1)^(n-1))(x^n)/n

In order to approximate ln(1.6) to within 0.0001, we need to find the least n where |pn(0.6) - ln(1.6)| < 0.0001.

Now, we can calculate pn(0.6) for different values of n until the condition is satisfied.