Taylor polynomial of f(x) = 1/(1-x) and the estimate of its remainder

[Ryker]

Homework Statement

Find the Taylor polynomial for f(x) = 1/(1-x), n = 5, centered around 0. Give an estimate of its remainder.

The Attempt at a Solution

I found the polynomial to be 1 + x + x² + x³ + x⁴ + x⁵, and then tried to take the Lagrange form of the remainder, say, for x in [-1/2, 1/2].

Then I get
$$R_{5}(x) = \frac{f^{6}(a)}{6!}x^{6}$$

But
$$f^{6}(a) = \frac{720}{(1-a)^{7}},$$
so for the largest values of a and x, that is 1/2, you get that
$$|R_{5}(x)| \leq \frac{(\frac{1}{2})^{6}}{(\frac{1}{2})^{7}} = 2,$$
which means the best estimate we can give is that the remainder is less than 2. But that doesn't make much sense to me, because the 5-th degree polynomial is fairly accurate on [-1/2, 1/2] and really close to f(x), so how I am getting such results? Have I made a mistake somewhere?

 

[Dick]

That looks fine to me. The remainder 'estimate' isn't really an estimate of the remainder. It's a maximum upper bound for the remainder. In this case it's a pretty pessimistic estimate, probably because you're nearing the singularity at x=1.

 

[Ryker]

Hey, thanks for the reply. Yeah, I figured it's only the maximum upper bound, so should I have taken a smaller interval? I mean, we aren't really given any specific range, so I guess we can choose our own, it's just that I didn't want to choose one, smaller than [-1/2, 1/2]. Or am I right in thinking none of this really matters, as long as you show how you got to that estimate?

 

[Dick]

Hey, thanks for the reply. Yeah, I figured it's only the maximum upper bound, so should I have taken a smaller interval? I mean, we aren't really given any specific range, so I guess we can choose our own, it's just that I didn't want to choose one, smaller than [-1/2, 1/2]. Or am I right in thinking none of this really matters, as long as you show how you got to that estimate?

I don't see anything wrong with what you've got. Maybe they just want the expression for R_5, not a specific number.

 

[Ryker]

I put down both just in case anyway, so I guess I should be covered then Thanks again.