1. ## taylor polynomials question

For what values of x is the approximate formula: ln(1 + x) = x - (1/2)(x^2) correct to 3 decimal places?

so i know that this is basically the maclaurin series representation of ln(1 + x) since a = 0. i used the convergence test for alternating series and confirmed that this series is indeed converging. so therefore i use the formula for the remainder which is Rn < |a_(n+1)|.

so the first term of the series omited is (1/3)x^3, so |x^3| / 3 should be less than .001 right? since you want the error to be within 3 decimal places. but in my book, they did |x^3| / 3 < .0005. how did they get that?

2. Originally Posted by oblixps
For what values of x is the approximate formula: ln(1 + x) = x - (1/2)(x^2) correct to 3 decimal places?

so i know that this is basically the maclaurin series representation of ln(1 + x) since a = 0. i used the convergence test for alternating series and confirmed that this series is indeed converging. so therefore i use the formula for the remainder which is Rn < |a_(n+1)|.

so the first term of the series omited is (1/3)x^3, so |x^3| / 3 should be less than .001 right? since you want the error to be within 3 decimal places. but in my book, they did |x^3| / 3 < .0005. how did they get that?
1.0006 to three decimal place is 1.001 but the error is 0.0004

CB

3. Originally Posted by CaptainBlack
1.0006 to three decimal place is 1.001 but the error is 0.0004

CB
no you didn't answer my question. besides the question in my book isn't asking for the error. my question was:

the first omitted term is (1/3)x^3, so |x^3| / 3 should be less than .001 right? since you want the error to be within 3 decimal places.

but in my book, they did |x^3| / 3 < .0005. how did they get that?

4. I think Captain Black already answered your question. If you want an answer to be surely correct to 3 decimal places the maximum error has to be no more than half of 0.001 i.e. 0.0005 and not 0.001. If you use any larger value the answer will sometimes be correct but sometimes the last digit will be wrong by 1.