If a question asks to use a power series to approximate a given definite integral to six decimal places and my power series ends up being an alternating series, I would use the alternating series error bound to help find the answer. If $|R_n|=|($actual sum of series$)-($partial sum for the nth term$)|$, would I write $|R_n|<0.000001$ or $|R_n|\leq0.000001$? Also, I've just memorized the fact that, for the phrase "to n decimal places," you write a decimal with 5 zeros followed by a 1 after the decimal point. But intuitively, why is the upper bound of the absolute error not $0.0000001$ or $0.00001$ with six and four zeros after the decimal place respectively?
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I think this answer might be of help. – Fimpellizzeri Nov 09 '19 at 05:05
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@Fimpellizieri One of the answers in the link you provided says that "to $n$ decimal places" means that the $n$th decimal digit and all preceding digits of the estimation are correct. This happens when the error is at most $10^{-n}$. Is this correct? I took the number $0.1254377$ and if I want an estimation correct to six decimal places, I can do $0.1254377-10^{-6}$ and I should get an answer correct to six decimal places. However, I get $0.1254367$ which is correct to not six but five decimal places. – Nov 09 '19 at 05:52
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I think when $n$-th decimal digit and all preceding digits of the estimation are correct, in principle you may need arbitrarily accurate estimates. Consider $1 - 10^{-n} = 0.\underbrace{99\dots9}_{n\text{ nines}}$. If the error is at least $10^{-n}$, then we may end with all decimal digits wrong, regardless of precision sought. Without knowing the digits that come after, it's hard to know how precise you must be in this case. – Fimpellizzeri Nov 09 '19 at 15:03