Question

In Section \(8.3,\) we established that the geometric series \(\sum r^{k}\) converges provided \(|r| < 1\). Notice that if \(-1 < r<0,\) the geometric series is also an alternating series. Use the Alternating Series Test to show that for \(-1 < r <0\), the series \(\sum r^{k}\) converges.

Short answer

Question: Prove that the geometric series \(\sum r^{k}\) converges for \(-1 < r <0\). Answer: For the given geometric series \(\sum r^{k}\), since the terms are alternating, non-increasing, and their limit approaches zero as \(k \to \infty\), the series converges for \(-1 < r < 0\) by the Alternating Series Test.

Step-by-step solution

Since \(r\) is negative, the series is alternating, and can be rewritten as \(\sum (-1)^k |r|^k\), where \(|r|\) is positive since \(-1 < r < 0\). Now, let \(a_k = |r|^k\). #Step 2: Show that terms are non-increasing#

We want to prove that \(a_{k+1} \leq a_k\) for all \(k\). Since \(|r| < 1\), it is clear that \(|r^{k+1}| = |r|^k \cdot |r| \leq |r|^k\). Thus, the terms in the series are non-increasing. #Step 3: Find the limit as k approaches infinity#

We need to find the limit of \(a_k\) as \(k \to \infty\). So, we compute the limit: \(\lim_{k\to\infty} |r|^k = \lim_{k\to\infty} \exp(k\log|r|)\). Since \(-1 < r < 0\), then \(0 < |r| < 1\) and \(\log|r| < 0\). Thus, as \(k \to \infty\), \(k\log|r| \to -\infty\) so \(\lim_{k\to\infty} \exp(k\log|r|) = \lim_{k\to\infty} |r|^k = 0\). #Step 4: Conclusion#

Since the series is alternating, the terms are non-increasing, and the limit of the sequence is zero, we can use the Alternating Series Test to conclude that the geometric series \(\sum r^{k}\) converges for \(-1 < r <0\).

Key concepts

Convergence of Series

In mathematics, a series is a sum of terms that follows a specific sequence. Understanding whether a series converges, meaning that it approaches a finite limit, is an important concept. One critical aspect when analyzing whether a series converges is to examine the behavior of its terms as they are added. This involves looking at the terms' size and behavior as the series progresses.

For example, if the terms of a series become smaller and approach zero, there’s a good chance the series will converge. An important tool for proving convergence is the **Alternating Series Test**, which is often used when the series alternates in sign, meaning the terms are positive in some steps and negative in others.

Ultimately, if the partial sums of a series (the totals computed as consecutive terms are added) approach a particular value, it shows that the series converges to that value.

Geometric Series

A geometric series is a series with a constant ratio between successive terms. Typically, it looks like this: \(\sum_{k=0}^{\infty} ar^k \) where \(a\) is the first term, and \(r\) is the common ratio.

Geometric series have special properties. They converge when the absolute value of the common ratio \(|r| < 1\). This means as you keep adding more terms, the sum doesn't go to infinity but instead settles at a particular number. For instance, the series \(\sum_{k=0}^{\infty} r^k \) converges to the sum \(\frac{1}{1-r} \) when \(|r| < 1\).

When the common ratio is negative, specifically in the range \(-1 < r < 0\), it means that the series also becomes an alternating series. Geometric series are easy to handle mathematically due to their regular and predictable pattern of term changes.

Alternating Series

Alternating series are series whose terms alternate in sign. A typical form of an alternating series looks like: \(\sum_{k=0}^{\infty} (-1)^k b_k \).

The **Alternating Series Test** is crucial to determine convergence of these series. According to this test, an alternating series will converge if two conditions are met:

• The absolute value of the terms decreases steadily, meaning each term is smaller than or equal to the one before it, \( b_{k+1} \leq b_k \).
• The limit of the term’s absolute value approaches zero as \(k\) goes to infinity: \( \lim_{k\to\infty} b_k = 0 \).

For the series in the original exercise, with \(-1 < r < 0\), it can reshape like an alternating series, confirming convergence through this test. Using these criteria ensures that each new term added reduces the total impact more than the previous one, leading to an overall reach towards a fixed sum.