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## 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 + x

^{2}+ x

^{3}+ x

^{4}+ x

^{5}, and then tried to take the Lagrange form of the remainder, say, for x in [-1/2, 1/2].

Then I get

[tex]R_{5}(x) = \frac{f^{6}(a)}{6!}x^{6}[/tex]

But

[tex]f^{6}(a) = \frac{720}{(1-a)^{7}},[/tex]

so for the largest values of a and x, that is 1/2, you get that

[tex]|R_{5}(x)| \leq \frac{(\frac{1}{2})^{6}}{(\frac{1}{2})^{7}} = 2,[/tex]

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?