convergence divergence sequence and series worksheet pdf

Convergence and Divergence of Sequences

A sequence is simply an ordered list of elements, often numbers, generated by a specific rule. Determining whether a sequence converges or diverges involves analyzing its behavior as the index approaches infinity.

Definition of Convergence and Divergence for Sequences

A sequence {a_n} is said to converge to a limit L if, for every ε > 0, there exists a positive integer N such that |a_n ⎻ L| < ε for all n > N. Intuitively, this means that the terms of the sequence get arbitrarily close to L as n becomes sufficiently large.

Formally, the limit of the sequence exists and is equal to L, denoted as lim_n→∞ a_n = L. If a sequence does not converge, it is said to diverge. Divergence can occur in several ways: the sequence may oscillate, grow without bound (approach infinity), or approach different values depending on the subsequence considered. The terms of a divergent sequence do not approach a single, finite value.

Convergence and Divergence of Series

An infinite series is the sum of an infinite sequence of terms. Determining whether a series converges or diverges involves examining the behavior of its partial sums as the number of terms increases without bound.

Definition of Convergence and Divergence for Series

An infinite series, represented as Σ a_n (where ‘n’ ranges from 1 to infinity), is said to converge if the sequence of its partial sums converges to a finite limit. The nth partial sum, denoted as S_n, is the sum of the first ‘n’ terms of the series: S_n = a₁ + a₂ + … + a_n.

Formally, the series converges if lim (as n approaches infinity) of S_n exists and is equal to a finite number, ‘L’. In this case, we say that the series converges to ‘L’, and we write Σ a_n = L. Conversely, if the sequence of partial sums does not converge (i.e., it either oscillates, approaches infinity, or approaches negative infinity), then the series is said to diverge. Divergence implies that the sum does not approach a finite value.

Tests for Convergence and Divergence of Series

Several tests exist to determine the convergence or divergence of infinite series. These tests provide methods for analyzing the behavior of series based on their terms.

The Divergence Test

The Divergence Test, also known as the nth-Term Test, is a preliminary check for the divergence of an infinite series. It’s straightforward: if the limit of the series’ terms does not approach zero as n approaches infinity, the series diverges.

Formally, if lim (n→∞) a_n ≠ 0, then the series Σ a_n diverges. This test is inconclusive if the limit equals zero; it doesn’t guarantee convergence. The Divergence Test should be your first resort for a quick divergence check.

For instance, consider the series Σ (n / (n + 1)). As n approaches infinity, the terms approach 1, not 0. Therefore, by the Divergence Test, this series diverges. Conversely, if lim (n→∞) a_n = 0, other convergence tests are needed to determine if the series converges.

Geometric Series Test

A geometric series is a series where each term is multiplied by a constant ratio ‘r’ to obtain the next term. Its general form is Σ arⁿ, where ‘a’ is the first term and ‘r’ is the common ratio. The Geometric Series Test provides a simple criterion for determining convergence or divergence based on the value of ‘r’.

Specifically, a geometric series converges if |r| < 1 and diverges if |r| ≥ 1. When it converges, the sum can be calculated using the formula S = a / (1 ⎻ r). This test is powerful due to its straightforward application.

For example, the series Σ (1/2)ⁿ is geometric with r = 1/2. Since |1/2| < 1, the series converges to 1 / (1 ⎻ 1/2) = 2. On the other hand, Σ 2ⁿ diverges because |2| ≥ 1.

Comparison Test

The Comparison Test is a method used to determine the convergence or divergence of a series by comparing it to another series whose convergence or divergence is already known. This test is particularly useful when dealing with series that resemble known convergent or divergent series.

The basic idea is as follows: Suppose we have two series, Σa_n and Σb_n, with positive terms. If Σb_n converges and a_n ≤ b_n for all n, then Σa_n also converges. Conversely, if Σb_n diverges and a_n ≥ b_n for all n, then Σa_n also diverges.

The success of the Comparison Test hinges on choosing an appropriate series Σb_n to compare with. Often, p-series or geometric series are used as benchmarks due to their well-defined convergence properties.

Limit Comparison Test

The Limit Comparison Test provides another way to determine the convergence or divergence of a series by comparing it to another series with known behavior. Unlike the direct Comparison Test, this test focuses on the limit of the ratio of the terms of the two series.

Consider two series, Σa_n and Σb_n, with positive terms. If the limit as n approaches infinity of (a_n / b_n) equals a finite number greater than zero, then either both series converge or both series diverge.

In essence, if the terms of the two series become proportional to each other as n gets very large, they will have the same convergence behavior. This test is often easier to apply than the direct Comparison Test, as it does not require finding an inequality between the terms of the series.

Ratio Test

The Ratio Test is a powerful tool for determining the convergence or divergence of series, especially those involving factorials or exponential terms. It examines the ratio of consecutive terms in the series to assess its behavior as n approaches infinity.

Given a series Σa_n, calculate the limit as n approaches infinity of |a_n+1 / a_n| = L. If L < 1, the series converges absolutely. If L > 1 or L = ∞, the series diverges. If L = 1, the test is inconclusive, and another test must be used.

The Ratio Test is particularly effective when dealing with series where the terms involve products or quotients that simplify when taking the ratio of consecutive terms. However, it may be inconclusive for certain series, such as those involving only polynomial terms.

Strategies for Determining Convergence or Divergence

Successfully determining convergence or divergence involves recognizing series types and strategically choosing the appropriate test. Practice is key to mastering these techniques and efficiently analyzing infinite series.

Recognizing Series Types

Identifying the type of series is the first crucial step in determining its convergence or divergence. Some common series types include geometric series, p-series, and telescoping series. Geometric series have a constant ratio between successive terms, making the geometric series test applicable.

P-series are of the form 1/n^p, where p is a real number, and their convergence depends on the value of p. Telescoping series involve terms that cancel out, simplifying the series to a finite sum. Recognizing these types allows for the application of specific tests tailored to their characteristics.

Furthermore, understanding the structure of the series, such as alternating signs or factorials, can guide the selection of appropriate tests like the alternating series test or the ratio test, respectively. Skillful recognition of series types significantly streamlines the convergence determination process.

Choosing the Appropriate Test

Selecting the correct test for convergence or divergence is paramount. Begin by considering the series’ form. If it resembles a geometric series, the geometric series test is appropriate. For series with terms approaching zero, the divergence test might be useful, though it only proves divergence.

When dealing with series involving factorials or exponential terms, the ratio test often proves effective. Comparison tests, including the direct comparison test and the limit comparison test, work well for series that can be compared to known convergent or divergent series, such as p-series.

The integral test applies to series with terms that correspond to a continuous, positive, and decreasing function. Alternating series can be tackled using the alternating series test. Remember that some tests are inconclusive under certain conditions, necessitating the use of alternative methods. Practice and familiarity with various tests are essential for efficient test selection.