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Chapter 9: Problem 76

Consider the following infinite series. a. Write out the first four terms of the sequence of partial sums. b. Estimate the limit of \(\left\\{S_{n}\right\\}\) or state that it does not exist. $$\sum_{k=1}^{\infty}(-1)^{k} k$$

Short Answer

Expert verified
A: The first four terms of the sequence of partial sums are: \(S_1 = -1, S_2 = 1, S_3 = -2, S_4 = 2\). Q: Does the limit of the sequence of partial sums exist for this alternating series? A: No, the limit of the sequence of partial sums does not exist, as the series diverges.

Step by step solution

01

Find the partial sum, \(S_n\)

In order to find the first four terms of the sequence of partial sums, we need to find the expression for \(S_n = \sum_{k=1}^{n} (-1)^k k\). The given series is: $$\sum_{k=1}^{\infty}(-1)^{k} k$$ So the partial sum \(S_n\) is: $$S_n = \sum_{k=1}^{n}(-1)^k k$$
02

Calculate the first four partial sums

Now, we will calculate the first four terms of the sequence of partial sums. \(S_1 = -1\) \(S_2 = -1 + 2\) \(S_3 = -1 + 2 - 3\) \(S_4 = -1 + 2 - 3 + 4\) which simplifies to: \(S_1 = -1\) \(S_2 = 1\) \(S_3 = -2\) \(S_4 = 2\)
03

Decide if the sequence converges or diverges

For an alternating series \(\sum_{k=1}^{\infty}(-1)^k a_k\) with \(a_k > 0, \forall k\) , we can apply the Leibniz test for convergence if: 1. The terms of the sequence \(a_k (k = 1,2,3, ...)\) are decreasing. 2. The limit \(\lim_{k \to \infty} a_k\) exists and is equal to 0. Our series converges if it satisfies the above conditions. In our case, the sequence of terms is: $$(-1)^k k = a_k$$ The sequence \( a_k=k\) is neither decreasing nor does the limit exist as: $$\lim_{k \to \infty} k = \infty \neq 0$$ Hence, the given alternating series does not satisfy the Leibniz test for convergence and hence diverges.
04

State the results

The sum of the first four terms of the sequence of partial sums is as follows: $$S_1 = -1, S_2 = 1, S_3 = -2, S_4 = 2$$ The limit of the sequence of partial sums, \(\left\\{S_{n}\right\\}\), does not exist as the series diverges. #Answer#: a. The first four terms of the sequence of partial sums are: \(S_1 = -1, S_2 = 1, S_3 = -2, S_4 = 2\) b. The limit of the sequence of partial sums, \(\left\\{S_{n}\right\\}\), does not exist, as the series diverges.

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Most popular questions from this chapter

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