Find a reasonable bound for error in approximating Taylor polynomial ln(1.5) about x = 0?

Question

Find a reasonable bound for error in approximating Taylor polynomial ln(1.5) about x = 0 ?

I need to know how to do it, please explain in detail.

Answer 1

As wriiten, this is extremely easy: ln(1.5) is a constant function, so its Taylor polynomial of every order is ln(1.5) and the error is 0.

The most plausible actual question that might have been misinterpreted as this seems to be:
Find a reasonable bound for the error in approximating ln(1.5) with the (nth?) Taylor polynomial for the function f(x) = ln (1+x) expanded about x = 0.

In this case the expansion is
Tn(x) = x - x^2/2 + x^3/3 - x^4/4 + ... + (-1)^(n+1) x^n/n.

The (n)th derivative of f(x) is (1+x)^(-n) . (-1)^(n+1) . (n-1)!
So the error term will be En = (1+z)^-(n+1) (-1)^(n+2) n! (0.5)^(n+1) / (n+1)! for some z between 0 and 0.5.
|En| = (1+z)^-(n+1) (0.5)^(n+1) / (n+1)
<= (0.5)^(n+1) / (n+1)
So this is an upper bound on the error.