Research Paper Abel and Dirichlet Tests for Convergence of Series

Abstract

The major focus of this study is to research the convergence of infinite series with Abel’s and Dirichlet's tests. These two simple tests offer tools to check for the convergence of some types of series, especially in cases when the terms have a specific pattern such as alternating terms or bounded monotonicity. The study introduces the formulation and application of both the wests, before it pros, cons and applications are compared. Also, the role of both tests is surveyed in the wider prize of series convergence. Besides, practical examples are examined to point out their strengths and weaknesses in different cases.

Keywords

Introduction

The convergence of infinite series is a core part of mathematical analysis, specifically in real analysis and complex analysis. Since many series that are encountered in different branches of mathematics do not satisfy the normal convergence criteria, it is important to develop specific tests for convergence. The most popular tools that belong to this class are the Abel Test and Dirichlet Test for convergence of series. Abel’s Test applies to series such as those that are bounded and are monotonically decreasing. Dirichlet’s Test is an assistant for a series which parts are a streaming part bounded function and a totally controllable oscillation, respectively.

In this paper the two tests have been put under the microscope, a deep study on both methods, and their theoretical foundations are included as well as their applications in the convergence process.

2.Abel’s Test for Convergence of Series

2.1 Statement of Abel’s Test

Abel's Test is cautiously used for series in the form of:

∑n=1∞anbn\sum_{n=1}^{\infty}a_n b_nn=1∑∞anbn

where:

• ana_nan is a monotonic and bounded sequence,

• bnb_nbn is a sequence of partial sums of some bounded sequence, meaning {bn}\{ b_n \}{bn} is bounded.

Abel's Test states that the series ∑n=1∞anbn\sum_{n=1}^{\infty} a_n b_n∑n=1∞anbn converges if:

• {an}\{a_n\}{an} is monotonic and bounded,
• {bn}\{b_n\}{bn} is bounded.

2.2 Proof of Abel’s Test

Let SN=∑n=1NbnS_N = \sum_{n=1}^{N} b_nSN=∑n=1Nbn be the partial sums of the series and suppose that

{bn}\{ b_n\}{bn} can be concatenated to some constant MMM. The series

∑n=1∞anbn\sum_{n=1}^{\infty} a_n b_n∑n=1∞anbn is said to converge if the limit Σ

(aαn)Σ\left(a_{\alpha}^{n}\right)Σ(aαn)n→∞ of the sequence {anSn}\{ a_n S_n \}{anSn} to zero is achieved as n approaches infinity.

In more formal terms, in order to prove the result, either one can choose the way of integral by parts calculation or think of the series as already being represented in terms of the sum of integrals and then come up with the fact that its convergence will be achieved under those conditions. It should be noted that the test is based on the idea that the monotonity of ana_nan and the boundedness of their product with the sum of the partial sums affect the series growth, while the boundedness of SnS_nSn impedes the oscillation of bnb_nbn too much.

2.3 Applications of Abel’s Test

The Abel’s Test which is what is only use to determine the truth of sequences said to be consistent when the items ana_nan are arranged in the order of decreasing geometric progression (GP) and when the Ss are such that the insertional is the mean of two A_n. The first term S_0 in the series and the last term S_n are in the order of decreasing exponential progression (EP). The nth term Sn itself and its position n speak of the general nature of the series itself.

∑n=1∞(−1

3. Dirichlet’s Test for Convergence of Series

3.1 Statement of Dirichlet’s Test

Dirichlet’s Test is the topic discussed when we have series such as:

∑n=1∞anbn\sum_{n=1}^{\infty} a_n

b_nn=1∑∞anbn where:

{an}\{a_n\}{an} is a monotonic sequence and it is bounded and {bn}\{b_n\}{bn} is a sequence with bounded partial sums, i.e., the sequence of partial sums SN=∑n=1NbnS_N = \sum_{n=1}^{N} b_nSN =∑n=1Nbn is bounded.

Dirichlet’s Theorem suggests that if the sequence {an}\{a_n\}{an} happens to be monotonic and bounded, while the partial sums SN

3.2 Proof of Dirichlet’s Test

The major idea behind Dirichlet's test is that the partialtlsums of bnb_nbn would be limited, therefore the oscillations would not be allowed todrunarbitrarily big. To be more exact, this test applies the summation by parts method where a series is expressed throughn sumsn resulting in the bouded difference between the balance of the seriesterms, as nnn grows. The finiteness of the partial sums of bnb_nbn will guarantee convergence of the sum of the series anbna_n b_nanbn that does not grow infinite.

3.3 Applications of Dirichlet’s Test

Dirichlet's Test is a technique that is mostly used in cases where ana_nan is a monotonic sequence and bnb_nbn is an oscillating sequence. A typical usage is on the alternating series whose one term slowly alters.

∑n=1∞(−1)nn\sum_{n=1}^{\infty}

\frac{(-1)^n}{n}n=1∑∞n(−1)n

Thus,

an=1na_n= 111n, is bounded and monotonic, and bn=(−1)nb_n=(-1)^nbn=(−1)n is whose partial sums are bounded by 1, therefore, Dirichlet’s Test asserts the convergence of the series.

4. Comparison of Abel and Dirichlet Tests

Abel's and Dirichlet's tests, though useful as conditions for the convergence of series, are often not usable in the same situation. It actually depends on the problem: When one of the series is monotonic and the other is the sequence of the partial sums of a bounded sequence, for instance, in the Fourier series, the Abel's Test becomes very useful. On the other hand, the application of Dirichlet's Test comes in the situation when one sequence is either monotonic and bounded and the other one is just a wiggly curve that is decaying. Similarly, these tests are similar but also different themselves. However, they both have to be related due to the reason that they both exhibit conditions like monotonicity and boundedness in sequences to attain the convergence, and they can be combined in applications.

5. Practical Examples

5.1 Example 1: Applying Abel’s Test

Consider the series:

∑n=1∞(−1)nnlog (n)\sum_{n=1}^{\infty} \frac{(-1)^n}{n \log(n)}n=1∑∞nlog(n)(−1)n

Here, an=1nlog (n)a_n = \frac{1}{n \log(n)}an=nlog(n)1 is monotonically decreasing and tends to zero, and bn=(−1)nb_n = (-1)^nbn=(−1)n is bounded. By Abel’s Test, the series converges.

5.2 Example 2: Applying Dirichlet’s Test

Consider the series:

∑n=1∞(−1)nn\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}n=1∑∞n(−1)n

Here, an=1na_n = \frac{1}{\sqrt{n}}an=n1 is monotonic and bounded, and bn=(−1)nb_n = (-1)^nbn=(−1)n has partial sums bounded by 1. By Dirichlet’s Test, the series converges.

6. CONCLUSION

The Abel and Dirichlet tests are very useful tools for analyzing the convergence of infinite series, especially with alternating or oscillatory patterns. Through practical experience and applying these tests, one is able to prove a series converges accurately that would otherwise be difficult to analyze by employing standard convergence tests. These techniques are integral in areas of mathematics such as Fourier analysis, number theory, and mathematical physics. In each test the interaction between monotonicity, boundedness, and the behavior of partial sums is clearly demonstrated, thereby, revealing in what way these series properties control the general convergence of the given series. Future research might deal with the application of these tests to series containing functions with complex values or to more general functional spaces.

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