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.单项选择题

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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

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  

Let's have a look at the alternating geometric series \[\sum_{n=0}^{\infty}\left (-\frac{1}{2}\right)^{n}.\] What's the alternating series estimate for the difference between the sum of these series and \[1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}?\] (A positive fraction, input as a/b in lowest terms.)

Let ∑ 𝑛 = 1 ∞ ( − 1 ) 𝑛 𝑎 𝑛  be a series where ∀ 𝑛 ,   𝑎 𝑛 > 0   ∀ 𝑛 ,   𝑎 𝑛 > 𝑎 𝑛 + 1   lim 𝑛 → ∞ 𝑎 𝑛 = 0   Define  𝑆 = lim 𝑘 → ∞ 𝑆 𝑘   where    𝑆 𝑘 = ∑ 𝑛 = 1 𝑘 ( − 1 ) 𝑛 𝑎 𝑛 . Given 𝑆 5 = 3.5 ,   𝑆 6 = 3.6 ,   𝑆 7 = 3.56 ,   𝑆 8 = 3.58 .  Which of the following statements MUST be true? Select all the correct answers.

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