[size=150]The convergence/divergence test is a method used to determine whether a given sequence converges or diverges. It helps analyze the behavior of the terms in a sequence and provides insights into its long-term behavior.[br][br]There are various convergence/divergence tests available, each with its own conditions and applicability. Let's discuss some commonly used tests:[br][br]1. Limit Test:[br]The limit test states that if the limit of a sequence, as n approaches infinity, exists and is finite, then the sequence converges. Mathematically, if lim(n→∞) an = L (where "an" represents the terms of the sequence), and L is a finite number, then the sequence converges. If the limit does not exist or is infinite, the sequence diverges.[br][br]2. Comparison Test:[br]The comparison test involves comparing a given sequence with another known sequence to determine convergence or divergence. Suppose we have two sequences, "an" and "bn," and we know that bn is convergent. If an ≤ bn for all n, then if bn converges, an also converges. Similarly, if bn diverges, an also diverges.[br][br]3. Ratio Test:[br]The ratio test examines the ratio of consecutive terms in a sequence. If lim(n→∞) |an+1/an| = L < 1, where "an" represents the terms of the sequence, then the sequence converges. If L > 1 or if the limit does not exist, the sequence diverges.[br][br]4. Root Test:[br]The root test analyzes the n-th root of the absolute value of each term in a sequence. If lim(n→∞) |an|^(1/n) = L < 1, where "an" represents the terms of the sequence, then the sequence converges. If L > 1 or if the limit does not exist, the sequence diverges.[br][br]5. Integral Test:[br]The integral test relates the convergence of a sequence to the convergence of an associated function. Consider a sequence "an" and the corresponding function f(x). If f(x) is continuous, positive, and decreasing for x ≥ 1, and the integral of f(x) from 1 to infinity converges, then the sequence "an" also converges. If the integral diverges, then the sequence diverges.[br][br]It's important to note that these tests provide guidelines and conditions for convergence or divergence, but they may not always yield conclusive results for every sequence. Sometimes a sequence may not satisfy the conditions of a specific test, requiring the use of other techniques or approaches for analysis.[br][br]By applying these mathematical tests, we can assess the convergence or divergence of a sequence and gain insights into its long-term behavior.[/size]

[size=150]Here are examples illustrating the convergence/divergence tests using mathematical formulas:[br][br]1. Limit Test:[br]Consider the sequence an = 1/n. To determine its convergence, we evaluate the limit as n approaches infinity: lim(n→∞) (1/n) = 0. Since the limit is finite (0), the sequence converges.[br][br]2. Comparison Test:[br]Let's consider the sequence an = (3n[sup]2[/sup] + 2n)/(4n[sup]2[/sup] + n + 1). We compare it with the sequence bn = (3n[sup]2[/sup])/(4n[sup]2[/sup]) = 3/4. Since 0 ≤ an ≤ b[sub]n[/sub] for all n, and b[sub]n[/sub] = 3/4 is convergent, we conclude that the sequence an also converges.[br][br]3. Ratio Test:[br]Take the sequence an = (n[sup]2[/sup])/(2[sup]n[/sup]). We apply the ratio test: lim(n→∞) |(n+1)[sup]2[/sup]/(2[sup](n+1)[/sup] )/ |(n[sup]2[/sup])/(2[sup]n[/sup])| = 1/2. Since the limit is less than 1, the sequence converges.[br][br]4. Root Test:[br]Consider the sequence an = (3^n)/(n!). We apply the root test: lim(n→∞) |(3[sup]n[/sup])[sup](1/n)[/sup]/ (n!)[sup](1/n)[/sup]= 3/e. Since the limit is less than 1, the sequence converges.[br][br]5. Integral Test:[br]Let's examine the sequence an = 1/n. We associate it with the function f(x) = 1/x. The function is continuous, positive, and decreasing for x ≥ 1. By evaluating the integral of f(x) from 1 to infinity, we have ∫(1 to ∞) 1/x dx = ln(x) |(1 to ∞) = ln(∞) - ln(1) = ∞. Since the integral diverges, the sequence an also diverges.[br][br]These examples illustrate the application of convergence/divergence tests to different sequences. It's important to carefully analyze the conditions of each test and apply them appropriately to draw conclusions about the convergence or divergence of a given sequence.[/size]