Comparing with the Harmonic Series

Oct 13, 2025 by talynt.org 18 views

Does this series with alternating elements converge: $\sum_{k=10}^{+\infty}\frac{(-1)^k}{k+(-1)^k}$ ?

In the realm of mathematics, particularly in the field of calculus, sequences and series play a vital role in understanding various mathematical concepts. A series is a sum of the terms of a sequence, and its convergence or divergence is a crucial aspect of analysis. In this article, we will investigate the convergence of a specific series with alternating elements, given by the expression $\sum_{k=10}^{+\infty}\frac{(-1)^k}{k+(-1)^k}$ . We will delve into the properties of this series, analyze its behavior, and determine whether it converges or diverges.

The given series is an alternating series, where the sign of each term alternates between positive and negative. The general term of the series is given by $\frac{(-1)^k}{k+(-1)^k}$ , where $k$ starts from $10$ and goes to infinity. To understand the behavior of this series, let's break down the general term into two parts: the numerator and the denominator.

The numerator is $(-1)^k$ , which alternates between $1$ and $-1$ as $k$ increases. The denominator is $k+(-1)^k$ , which also alternates between two values: $k+1$ when $k$ is even and $k-1$ when $k$ is odd.

Analyzing the Denominator

To analyze the behavior of the series, we need to examine the denominator, which is $k+(-1)^k$ . When $k$ is even, the denominator becomes $k+1$ , and when $k$ is odd, the denominator becomes $k-1$ . This means that the denominator is always greater than or equal to $k-1$ .

Comparing with the Harmonic Series

The given series is similar to the harmonic series, which is known to diverge. However, the presence of the alternating term $(-1)^k$ in the numerator makes it different from the harmonic series. To understand the behavior of this series, let's compare it with the harmonic series.

The harmonic series is given by $\sum_{k=1}^{+\infty}\frac{1}{k}$ , and it is known to diverge. However, the given series has an additional term $(-1)^k$ in the numerator, which alternates between $1$ and $-1$ . This means that the terms of the given series are not as large as the terms of the harmonic series.

Testing for Convergence

To determine whether the given series converges or diverges, we need to test it for convergence. One way to do this is to use the Alternating Series Test, which states that an alternating series converges if the following conditions are met:

The terms of the series alternate in sign.
The absolute value of each term is less than or equal to the absolute value of the previous term.
The limit of the terms as $k$ approaches infinity is zero.

Let's apply these conditions to the given series.

Condition 1: Alternating Terms

The terms of the series alternate in sign, as the numerator $(-1)^k$ alternates between $1$ and $-1$ . This satisfies the first condition of the Alternating Series Test.

Condition 2: Decreasing Terms

To check if the absolute value of each term is less than or equal to the absolute value of the previous term, we need to examine the behavior of the denominator. As $k$ increases, the denominator $k+(-1)^k$ also increases. This means that the absolute value of each term is less than or equal to the absolute value of the previous term.

Condition 3: Limit of Terms

To check if the limit of the terms as $k$ approaches infinity is zero, we need to examine the behavior of the general term as $k$ approaches infinity. As $k$ approaches infinity, the denominator $k+(-1)^k$ also approaches infinity. This means that the limit of the terms as $k$ approaches infinity is not zero.

Based on the analysis above, we can conclude that the given series does not satisfy the conditions of the Alternating Series Test. Specifically, the limit of the terms as $k$ approaches infinity is not zero. This means that the series does not converge.

However, we can still investigate the behavior of the series further. One way to do this is to use the Leibniz Test, which states that an alternating series converges if the following conditions are met:

The terms of the series alternate in sign.
The absolute value of each term is less than or equal to the absolute value of the previous term.
The limit of the terms as $k$ approaches infinity is zero.

Let's apply these conditions to the given series.

Condition 1: Alternating Terms

The terms of the series alternate in sign, as the numerator $(-1)^k$ alternates between $1$ and $-1$ . This satisfies the first condition of the Leibniz Test.

Condition 2: Decreasing Terms

Condition 3: Limit of Terms

Based on the analysis above, we can conclude that the given series does not satisfy the conditions of the Leibniz Test. Specifically, the limit of the terms as $k$ approaches infinity is not zero. This means that the series does not converge.

However, we can still investigate the behavior of the series further. One way to do this is to use the Integral Test, which states that a series converges if the following conditions are met:

The terms of the series are positive.
The function $f(x)$ is continuous and positive on the interval $[1, \infty)$ .
The function $f(x)$ is decreasing on the interval $[1, \infty)$ .

Let's apply these conditions to the given series.

Condition 1: Positive Terms

The terms of the series are positive, as the numerator $(-1)^k$ alternates between $1$ and $-1$ . This satisfies the first condition of the Integral Test.

Condition 2: Continuous Function

To check if the function $f(x)$ is continuous and positive on the interval $[1, \infty)$ , we need to examine the behavior of the general term as $x$ approaches infinity. As $x$ approaches infinity, the denominator $x+(-1)^x$ also approaches infinity. This means that the function $f(x)$ is continuous and positive on the interval $[1, \infty)$ .

Condition 3: Decreasing Function

To check if the function $f(x)$ is decreasing on the interval $[1, \infty)$ , we need to examine the behavior of the general term as $x$ approaches infinity. As $x$ approaches infinity, the denominator $x+(-1)^x$ also approaches infinity. This means that the function $f(x)$ is decreasing on the interval $[1, \infty)$ .

Based on the analysis above, we can conclude that the given series satisfies the conditions of the Integral Test. Specifically, the terms of the series are positive, the function $f(x)$ is continuous and positive on the interval $[1, \infty)$ , and the function $f(x)$ is decreasing on the interval $[1, \infty)$ . This means that the series converges.

However, we can still investigate the behavior of the series further. One way to do this is to use the Comparison Test, which states that a series converges if the following conditions are met:

The terms of the series are positive.
The series $\sum_{k=1}^{+\infty}a_k$ converges.
The terms of the series $\sum_{k=1}^{+\infty}a_k$ are greater than or equal to the terms of the series $\sum_{k=1}^{+\infty}b_k$ .

Let's apply these conditions to the given series.

Condition 1: Positive Terms

The terms of the series are positive, as the numerator $(-1)^k$ alternates between $1$ and $-1$ . This satisfies the first condition of the Comparison Test.

Condition 2: Convergent Series

To check if the series $\sum_{k=1}^{+\infty}a_k$ converges, we need to examine the behavior of the general term as $k$ approaches infinity. As $k$ approaches infinity, the denominator $k+(-1)^k$ also approaches infinity. This means that the series $\sum_{k=1}^{+\infty}a_k$ converges.

**Q&A: Does this series with alternating elements converge: $\sum_{k=10}^{+\infty}\frac{(-1)^k}{k+(-1)^k}$?**

Q: What is the Alternating Series Test?

A: The Alternating Series Test is a test used to determine whether an alternating series converges or diverges. The test states that an alternating series converges if the following conditions are met:

The terms of the series alternate in sign.
The absolute value of each term is less than or equal to the absolute value of the previous term.
The limit of the terms as $k$ approaches infinity is zero.

Q: Does the given series satisfy the Alternating Series Test?

A: No, the given series does not satisfy the Alternating Series Test. Specifically, the limit of the terms as $k$ approaches infinity is not zero.

Q: What is the Leibniz Test?

A: The Leibniz Test is a test used to determine whether an alternating series converges or diverges. The test states that an alternating series converges if the following conditions are met:

The terms of the series alternate in sign.
The absolute value of each term is less than or equal to the absolute value of the previous term.
The limit of the terms as $k$ approaches infinity is zero.

Q: Does the given series satisfy the Leibniz Test?

A: No, the given series does not satisfy the Leibniz Test. Specifically, the limit of the terms as $k$ approaches infinity is not zero.

Q: What is the Integral Test?

A: The Integral Test is a test used to determine whether a series converges or diverges. The test states that a series converges if the following conditions are met:

The terms of the series are positive.
The function $f(x)$ is continuous and positive on the interval $[1, \infty)$ .
The function $f(x)$ is decreasing on the interval $[1, \infty)$ .

Q: Does the given series satisfy the Integral Test?

A: Yes, the given series satisfies the Integral Test. Specifically, the terms of the series are positive, the function $f(x)$ is continuous and positive on the interval $[1, \infty)$ , and the function $f(x)$ is decreasing on the interval $[1, \infty)$ .

Q: What is the Comparison Test?

A: The Comparison Test is a test used to determine whether a series converges or diverges. The test states that a series converges if the following conditions are met:

The terms of the series are positive.
The series $\sum_{k=1}^{+\infty}a_k$ converges.
The terms of the series $\sum_{k=1}^{+\infty}a_k$ are greater than or equal to the terms of the series $\sum_{k=1}^{+\infty}b_k$ .

Q: Does the given series satisfy the Comparison Test?

A: Yes, the given series satisfies the Comparison Test. Specifically, the terms of the series are positive, the series $\sum_{k=1}^{+\infty}a_k$ converges, and the terms of the series $\sum_{k=1}^{+\infty}a_k$ are greater than or equal to the terms of the series $\sum_{k=1}^{+\infty}b_k$ .

Q: What is the conclusion of the analysis?

A: Based on the analysis above, we can conclude that the given series converges. The series satisfies the conditions of the Integral Test and the Comparison Test, and it does not satisfy the conditions of the Alternating Series Test and the Leibniz Test.

Q: What are the implications of the conclusion?

A: The conclusion implies that the given series converges, and it has a finite sum. This means that the series can be approximated by a finite sum, and the error can be made arbitrarily small by increasing the number of terms.

Q: What are the limitations of the analysis?

A: The analysis is limited to the specific series given, and it does not provide a general result for all alternating series. The analysis relies on the specific properties of the given series, and it may not be applicable to other series with different properties.

Q: What are the future directions of research?

A: Future research directions may include investigating the convergence of other alternating series, developing new tests for convergence, and applying the results to real-world problems.