Question

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

Question: Write out the first four terms of the partial sums sequence, and determine if the limit of the sequence of partial sums {Sn} exists for the infinite series ∑ (-1)^k k. Answer: The first four terms of the sequence of partial sums are: S1 = -1, S2 = 1, S3 = -2, S4 = 2. Based on our analysis using the Alternating Series Test, the infinite series does not converge, and therefore, the limit of the sequence of partial sums {Sn} does not exist.

Step-by-step solution

Write the series and definition of partial sums

We are given the following infinite series: $$\sum_{k=1}^{\infty}(-1)^{k} k$$ The sequence of partial sums (\(S_n\)) is defined as: $$S_n = \sum_{k=1}^{n}(-1)^{k} k$$

Calculate the first four partial sums

To find the first four partial sums, we will calculate the sum of the first few terms of the given series: \(n=1\): \(S_1 = (-1)^1 1 = -1\) \(n=2\): \(S_2 = (-1)^1 1 + (-1)^2 2 = -1 + 2 = 1\) \(n=3\): \(S_3 = (-1)^1 1 + (-1)^2 2 + (-1)^3 3 = -1 + 2 - 3 = -2\) \(n=4\): \(S_4 = (-1)^1 1 + (-1)^2 2 + (-1)^3 3 + (-1)^4 4 = -1 + 2 - 3 + 4 = 2\)

Write out the first four terms of the sequence of partial sums

The first four terms of the sequence of partial sums are as follows: $$S_1 = -1, S_2 = 1, S_3 = -2, S_4 = 2$$

Analyze the sequence of partial sums

Since the infinite series alternates signs, we will use the Alternating Series Test to determine if the sequence converges. The general term of our series is \((-1)^k k\). Observe that the absolute value of successive terms decreases: $$|(-1)^1 1| > |(-1)^2 2| > |(-1)^3 3| > |(-1)^4 4|$$ However, the limit of the absolute value of the general term as \(k\) approaches infinity is not zero: $$\lim_{k\to\infty} |(-1)^k k| = \lim_{k\to\infty} k = \infty$$ According to the Alternating Series Test, if the limit of the absolute value of the general term is not zero, then the series does not converge.

Conclusion

Based on our analysis using the Alternating Series Test, the infinite series does not converge, and therefore, the limit of the sequence of partial sums \(\left\{S_n\right\}\) does not exist.

Key concepts

Partial Sums

When studying infinite series, a fundamental concept is understanding partial sums. These are the sums of the first few terms in a series, which are used to determine what value, if any, the series approaches as you add more and more terms. For any infinite series \[\sum_{k=1}^{\text{\infinity}} a_k\] the nth partial sum \(S_n\) is given by the finite sum: \[S_n = \sum_{k=1}^n a_k\].

Let's for example calculate the first few partial sums of the series \(\sum_{k=1}^{\text{\infinity}} (-1)^k k\). The sum of the first term, the first partial sum, would simply be \(S_1 = -1\). Add the second term, and the second partial sum becomes \(S_2 = -1 + 2\), and so on. In practice, these calculations allow us to peek at the behavior of the series' sum as we add more terms, which is crucial for understanding series convergence.

Alternating Series Test

The Alternating Series Test is a method used to determine the convergence of infinite series that alternate in sign. An alternating series is one in which each term is successively positive and negative, such as \(\sum (-1)^k a_k\), where \(a_k\) are positive terms. For the test to confirm that a series converges, two conditions must be met: firstly, the absolute values of the terms \(\|a_k\|\) must decrease with every step; secondly, the limit of these terms as \(k\) approaches infinity must be zero, \(\lim_{k\to\infty} a_k = 0\).

If we look at our example series \(\sum_{k=1}^{\text{\infinity}} (-1)^k k\), although the series alternates in sign, it fails the second condition, as the limit of \(k\) as \(k\) approaches infinity is not zero but rather grows without bound. According to this test, if even one of the conditions is not fulfilled, the series does not converge.

Sequence Convergence

Convergence in sequences is a central idea in calculus, indicating that the terms of the sequence get closer and closer to a specific value, known as the limit, as the sequence progresses. A sequence \(\{a_n\}\) converges to the number \(L\) if, for every positive number \(\epsilon\), there is a positive integer \(N\) such that for all \(n > N\), the absolute difference between \(a_n\) and \(L\) is less than \(\epsilon\): \(\|a_n - L\| < \epsilon\).

This concept is especially important when discussing infinite series, as the convergence of the sequence of partial sums is what indicates the convergence of the series itself. In the given exercise, due to the Alternating Series Test failing, we understand that the sequence of partial sums does not approach a finite limit, hence, it diverges.