Question

A series is given. (a) Find a formula for \(S_{n},\) the \(n^{\text {th }}\) partial sum of the series. (b) Determine whether the series converges or diverges. If it converges, state what it converges to. $$\sum_{n=1}^{\infty}(-1)^{n} n$$

Short answer

The series diverges as the partial sums increase in absolute value without settling.

Step-by-step solution

Understand the Series

The given series is \( \sum_{n=1}^{\infty}(-1)^{n} n \). Here, \((-1)^{n}\) is an alternating sign factor and \(n\) is a term from the natural numbers.

Investigate the Partial Sum

Let's denote the partial sum by \(S_n\), where \(S_n = \sum_{k=1}^{n} (-1)^{k} k \). We will calculate the first few terms to identify any pattern: \((-1)\), \((-1 + 2)\), \((-1 + 2 - 3)\), ... etc.

Pattern Recognition

Calculate explicitly first few partial sums: \(-1\), \(1\), \(-2\), \(2\),... Notice the partial sums alternate in sign and their absolute values increase without bound.

General Formula for Partial Sum

From our calculations, the pattern suggests that if \(n\) is even \(S_n = \frac{n}{2}\) and if \(n\) is odd \(S_n = -\frac{n+1}{2}\). This accounts for alternating signs and increasing magnitude.

Analyze Convergence or Divergence

Evaluate the behavior of the sequence of partial sums. Since the partial sums \(S_n\) alternate and continue to increase in absolute value as \(n\) increases, they do not settle to a finite limit.

Key concepts

Partial Sum Formula

Understanding the concept of partial sums is crucial in determining whether a series converges or diverges. A partial sum, represented as \(S_n\), is simply the sum of the first \(n\) terms of a given series. For the series \( \sum_{n=1}^{\infty}(-1)^{n} n \), the partial sum formula helps us see the series behavior more explicitly.

When looking for the partial sum, we examine the pattern in how these sums accumulate. This particular series alternates its sign due to the \((-1)^n\) factor, causing positive and negative terms to alternate. This results in:

• If \(n\) is even, the partial sum \(S_n = \frac{n}{2}\)
• If \(n\) is odd, the partial sum \(S_n = -\frac{n+1}{2}\)

These formulas arise from the pattern observed in the partial sums, where the absolute values rise while the signs alternate. Understanding this idea helps us see the nature of the series more clearly.

Alternating Series

The series given by \( \sum_{n=1}^{\infty}(-1)^{n} n \) is a classic example of an alternating series. Alternating series flip the sign of their terms, leading to potential cancellation effects that might help the series converge. The key characteristic is the presence of the \((-1)^n\) factor, which produces this alternating behavior.

However, the convergence of an alternating series isn't guaranteed just by its nature. We must consider the absolute value of the terms. For convergence, it's often needed that the absolute terms decrease in a specific manner, typically getting smaller and approaching zero. In our series, each term \(n\) grows instead of decaying, underscoring that the series diverges.

Therefore, while alternating series may have a convergent nature due to their cancelling parts, in this case, the aim of convergent behavior is defeated by the increasing terms.

Divergence Analysis

To determine if a series converges or diverges, it's essential to analyze the behavior of its partial sums. Divergence means that the partial sums do not approach a finite limit. In the case of the series \( \sum_{n=1}^{\infty}(-1)^{n} n \), even though the signs alternate, the partial sums were found to increase without bound.

Let’s recap the divergence analysis:

• The partial sums \(S_n = \frac{n}{2}\) for even \(n\) and \(S_n = -\frac{n+1}{2}\) for odd \(n\) indicate that, regardless of whether \(n\) is odd or even, the absolute value of \(S_n\) continues to grow.
• Since these sums do not stabilize and instead grow, they clearly illustrate the series' divergence.

Divergence analysis, therefore, is a critical approach to understand the overall behavior of a series, especially when simple intuition from alternating signs might suggest otherwise.