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. $$\sum_{k=1}^{\infty}(-1)^{k}$$

Short answer

Question: Determine the first four terms of the alternating series $$\sum_{k=1}^{\infty}(-1)^{k}$$ and analyze if it has a limit. Answer: The first four terms of the series are -1, 1, -1, 1. The sequence of partial sums alternates between -1 and 0, and does not approach a single value as the number of terms increases. Therefore, the infinite series does not have a limit.

Step-by-step solution

Identify the sequence

The given infinite series is an alternating series: $$\sum_{k=1}^{\infty}(-1)^{k}$$

Find the first four terms of the sequence

The first four terms of the sequence can be found by replacing k with 1, 2, 3, and 4 in the formula. We get: Term 1: \((-1)^1 = -1\) \\ Term 2: \((-1)^2 = 1\) \\ Term 3: \((-1)^3 = -1\) \\ Term 4: \((-1)^4 = 1\) \\ So, the sequence looks like: -1, 1, -1, 1, ...

Calculate the first four partial sums of the sequence

To find the first four partial sums of the sequence, we will get the sum of the terms of the sequence up to the nth term. Partial Sum 1: \(S_1 = -1\) \\ Partial Sum 2: \(S_2 = -1 + 1 = 0\) \\ Partial Sum 3: \(S_3 = -1 + 1 - 1 = -1\) \\ Partial Sum 4: \(S_4 = -1 + 1 - 1 + 1 = 0\) \\ So, the first four partial sums are -1, 0, -1, 0.

Determine the limit of the sequence of partial sums

To determine the limit of the infinite series, we observe the behavior of the sequence of partial sums as the number of terms increases. The sequence of partial sums seems to alternate between -1 and 0. As n goes to infinity, the sequence does not approach a single value but continues to alternate between -1 and 0.

State the conclusion

The limit of the sequence of partial sums does not exist due to the alternating nature of the sequence. Therefore, the infinite series $$\sum_{k=1}^{\infty}(-1)^{k}$$ does not have a limit as well.