determine the smallest number of terms required to approximate the sum of the convergent series with an error of less than 0.001.?
(a) Use the Alternating Series Remainder Theorem to determine the smallest number of terms required to approximate the sum of the convergent series with an error of less than 0.001.
- husoskiLv 74 months ago
The sign of the first omitted term gives the direction of the error and the magnitude of the error is less than the magnitude of that term. So, if you *truncate* that series at n terms, the first omitted term is (-1)^(n+2) / (n + 1). You need the magnitude of that term to be less than or equal to 0.001:
1/(n+1) <= 1/1000
1000 <= n + 1
n >= 999
So, you'd need 999 terms if all you know is that the series fits the alternating series pattern.
HOWEVER: in a very slowly converging alternating series (this one, the alternating harmonic series, is one of the very slowest!), the magnitude of the error is about half the omitted term. So, you get within 0.001 of ln(2) by summing only 500 terms in this case.