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3. Show that the approximation
e = 1 + 1 +1/2!+ · · · +1/7!
gives the value of e to within an error of 8 × 10−5.
Solution: The given approximation is the 7th-degree Maclaurin polynomial for ex evaluated at x = 1.
Since the 8th derivative of e^x is e^x, and the maximum (absolute) value of this 8th derivative on the
interval [0, 1] is e, the approximation has error at most e · (1 − 0)8/8! = e/40320. Since e < 3, the error
is < 3/40320, which is < 8 · 10−5 as required.
I get that the error is 1/8! but can anyone explain to me why we multiply e by this error to get the actual error? Or am i confused?
Thx
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