Math question with Taylor's remainder estimate !
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Math question with Taylor's remainder estimate !

[From: ] [author: ] [Date: 11-04-24] [Hit: ]
since |x| ≤ 1.= e/120.Note: If you want your answer without e in it,I hope this helps!......
According to Taylor's remainder estimate, the maximum possible error in the use of

4
Σ (x^n) / (n!)
n=0

to approximate e^x on the interval [-1,1] is which value?

-
The error is given by
|f '''''(c) x^5/5!| for some c between 0 and x.

Since f(x) = e^x, f '''''(x) = e^x.

So, |f '''''(c) x^5/5!|
= e^c |x|^5 / 5!
≤ e^c * 1^5 / 5!, since |x| ≤ 1.
< e^1 / 5!, since c < x ≤ 1
= e/120.

Note: If you want your answer without e in it, use a crude bound like e < 3
==> |Error| < 3/120 = 1/40.

I hope this helps!
1
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