Question

In Section 3, we established that the geometric series \(\Sigma r^{k}\) converges provided \(|r|<1\). Notice that if \(-1

Short answer

Question: Prove that the geometric series \(\sum r^{k}\) converges for \(-1<r<0\) using the Alternating Series Test. Answer: The series \(\sum r^{k}\) converges for \(-1<r<0\) because the sequence \(a_k = |r|^k\) is non-increasing, and the limit \(\lim_{k \to \infty} a_k = 0\), which satisfies the conditions of the Alternating Series Test.

Step-by-step solution

Review the Alternating Series Test

The Alternating Series Test states that an alternating series \(\sum (-1)^{k}a_k\) converges if both of the following conditions hold: 1. \(a_k\) is non-increasing, meaning \(a_{k+1} \leq a_k\) for all \(k\) 2. \(\lim_{k \to \infty} a_k = 0\) In our case, the given series is \(\sum r^{k}\) with \(-1 < r < 0\). We can rewrite this series as an alternating series by taking \(a_k = |r|^k\) and expressing the series as \(\sum (-1)^{k}a_k\).

Verify Condition 1 (non-increasing sequence) for the Alternating Series Test

To verify that the sequence \(a_k = |r|^k\) is non-increasing, we need to show that \(a_{k+1} \leq a_k\) for all \(k\). Given that \(-1 < r < 0\), we know that \(0 < |r| < 1\). Thus, for any \(k \geq 0\): $$a_{k+1} = |r|^{k+1} = |r|^{k} \cdot |r| = a_k \cdot |r|.$$ Because \(0 < |r| < 1\), we can conclude that \(a_{k+1} \leq a_k\) for all k.

Verify Condition 2 (limit) for the Alternating Series Test

To verify the second condition, we need to show that \(\lim_{k \to \infty} a_k = 0\). Here, \(a_k = |r|^k\). Since \(0 < |r| < 1\), when \(k\) goes to infinity, the value of \(|r|^k\) approaches 0. So, we have: $$\lim_{k \to \infty} a_k = \lim_{k \to \infty} |r|^k = 0.$$

Apply the Alternating Series Test

We have shown that both conditions of the Alternating Series Test hold for the geometric series \(\sum r^{k}\) with \(-1 < r < 0\): 1. The sequence \(a_k = |r|^k\) is non-increasing 2. The limit \(\lim_{k \to \infty} a_k = 0\) Therefore, according to the Alternating Series Test, the series \(\sum r^{k}\) converges when \(-1 < r < 0\).

Key concepts

Geometric Series

A geometric series is a series of the form \( \sum_{k=0}^{\infty} ar^k \) where \( a \) is the initial term and \( r \) is the common ratio. These types of series have a unique property: they converge or sum up to a finite number if the absolute value of the common ratio \( r \) is less than one, i.e., \(|r| < 1\).

If \(|r| \geq 1\), the series will diverge, meaning it will not settle on any finite value. Geometric series can be used in various fields, like calculating interest in finance or analyzing signal patterns in engineering.

Importantly, if \( -1 < r < 0 \), the series is not only a geometric series but also an alternating series. Alternating series change the sign for each term, which has a significant impact on their convergence properties.

Convergence

Convergence refers to the behavior of a series as more and more terms are added up. A convergent series approaches a definite value, no matter how many terms you add, while a divergent series does not settle to a specific number.

To determine if a series converges, there are different tests available, one being the Alternating Series Test. This specific test helps identify when a series that alternates in signs (positive, negative, positive, etc.) converges.

In the context of the geometric series with \(-1 < r < 0\), convergence can be confirmed by rewriting it as an alternating series and applying the Alternating Series Test. Since the series meets all necessary conditions of this test, it converges despite the alternating signs.

Non-Increasing Sequence

A sequence is non-increasing if each term is less than or equal to the previous term. Mathematically, for a sequence \(a_k\), this condition is expressed as \(a_{k+1} \leq a_k\).

For the series \(\sum r^k \) where \(-1 < r < 0\), we take \(a_k = |r|^k\). Because \(0 < |r| < 1\), multiplying each \(a_k\) by \(|r|\) results in a smaller number due to the multiplication by a fractional amount.

Verified through the Alternating Series Test, this non-increasing nature ensures that the sequence gradually decreases, supporting the idea of convergence of the series.

Limit

The concept of a limit is fundamental to understanding series and sequences in mathematics. A limit investigates what a sequence or function approaches as you go indefinitely along its path, such as to infinity.

For the geometric series \(\sum r^k \) with \(-1 < r < 0\), and \(a_k = |r|^k\), we find that as \(k\) approaches infinity, \(|r|^k\) gets closer to zero due to \(0 < |r| < 1\).

Proving the limit is zero is crucial for applying the Alternating Series Test, as it satisfies one of its primary conditions. The fulfillment of this condition confirms that the terms become negligible as they extend towards infinity, ensuring the convergence of the series.