Question textThe following alternating harmonic series is an approximation of [math: ln⁡(2)]: [math: ∑n=1∞(−1)n+1n] How many terms of the series are required such that the absolute difference between the series value and [math: ln⁡(2)] is less than 0.0005? Number of terms required to approximate [math: ln⁡(2)]: Answer 1 Question 3[input] Approximation of [math: ln⁡(2)] (round to 6 decimal places): Answer 2 Question 3[input]Check Question 3

Question

BlackTom AI · Accepted Answer

The alternating series estimation theorem gives that for an alternating series with terms decreasing in magnitude to 0, the absolute error after N terms is at most the magnitude of the next term. For the alternating harmonic series ∑_{n=1}∞ (−1)^{n+1}/n (which converges to ln 2), the remainder after N terms satisfies |ln 2 − S_N| ≤ 1/(N+1). Step‑by‑step reasoning to find N: 1. Require |ln 2 − S_N| < 0.0005, so enforce 1/(N+1) < 0.0005. 2. Solve 1/(N+1) < 0.0005 ⇒ N+1 > 1/0.0005 = 2000. 3. Thus N > 1999, so the smallest integer N satisfying this is N = 2000. Calculation and approximation of S_2000: 1. For even N (like 2000), S_N is slightly less than ln 2 and the remainder R_N = ln 2 − S_N is positive and small. 2. Asymptotically the remainder for large N behaves like 1/(2N) (from Euler–Maclaurin / asymptotic expansion of the alternating tail), so for N = 2000 we approximate R_2000 ≈ 1/(2·2000) = 1/4000 = 0.00025. 3. Therefore S_2000 ≈ ln 2 − 0.00025. Using ln 2 ≈ 0.69314718056, S_2000 ≈ 0.69314718056 − 0.00025 = 0.69289718056. Rounded to six decimal places this is 0.692897. Why the answers are correct: - N = 2000 is the minimum number of terms because N = 1999 would give the next-term bound 1/2000 = 0.0005, which does not satisfy the strict “less than 0.0005” requirement. - The numerical approximation 0.692897 follows from subtracting the estimated remainder ≈ 0.00025 from ln 2 and rounding; the true S_2000 lies between ln 2 − 1/2001 and ln 2, so the value 0.692897 is consistent and accurate to six decimals. Confirmed final answers: Number of terms = 2000; Approximation (6 d.p.) = 0.692897.

类似问题

Which of the following summations is equal to 1 − 1 3 + 1 5 − 1 7 + 1 9 − 1 11 + 1 13 − 1 15 + 1 17 − 1 19

Estimate the series [math: ∑c=1∞(−1)c+1c3] \displaystyle\sum\limits_{c=1}^ \infty \frac{(-1)^{c+1}}{c^3} using the first three terms of the series.

Use [math: ∑n=1∞1n2=π26]\displaystyle \sum\limits_{n=1}^ \infty \frac{1}{n^2} = \frac{\pi{^2}}{6} to find the sum of the series [math: ∑n=1∞(−1)n+1n2]\displaystyle \sum\limits_{n=1}^ \infty \frac{(-1)^{n+1}}{n^2} .

Which of the following trees corresponds to a potential parse of the ambiguous sentence below, with correct syntactic categories? Some diagnostics are provided.

Which of the following sentences contain two non-finite verbs? Select all that apply.

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单