- #1

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[tex]

f(x) = e^x

[/tex]

about x=0 and hence approximate the value of e. Estimate the error in the approximation.

I have written the taylor polynomial and evaluated for x=1 to give an approximation of e.

Its just the error that is confusing me. I have:

[tex]

R_n(x) = \frac{f^{(n+1)}(z)}{(n+1)!}(x-a)^{n+1} = \frac{e^z}{10!}

[/tex]

and I need to find an upper bound for this to give the maximum error of the approximation.

So far I have

[tex]

0 < z < 1, e^0 < e^z < e^1

[/tex]

and then

[tex]

\frac{1}{10!} < \frac{e^z}{10!} < \frac{e}{10!}

[/tex]

but then the upper bound has an e in it. In a similar example in the book they have just put in 3 instead of e, I guess making the assumption that e < 3. But I'm thinking if i do this I should somehow show that e < 3.

Thanks for any help :)