Use The Comparison Test To Determine Convergence Or Divergen

Use The Comparison Test To Determine The Convergence Or Divergence

Analyze a series by applying the comparison test to determine whether it converges or diverges. This involves selecting a known benchmark series that is comparable in behavior to the given series and then comparing their terms to draw conclusions about convergence.

Apply the nth term test to evaluate whether a series converges or diverges by examining its general term as n approaches infinity. If the limit of the term is not zero, then the series diverges.

Use the integral test on a series by integrating a related continuous function over an appropriate interval and comparing the integral's convergence with the series.

Apply the alternating series test to series with alternating positive and negative terms, checking the monotonicity and limit conditions for convergence.

Use the root test by taking the nth root of the absolute value of the terms, then evaluating the limit to determine the series' convergence or divergence.

Paper For Above instruction

Series analysis is a fundamental component of mathematical analysis, offering insights into the behavior of infinite sums. Several convergence tests exist, each suited to different types of series, such as the comparison test, nth term test, integral test, alternating series test, and root test. This paper explores these methods with practical applications and detailed reasoning.

A. Comparison Test

The comparison test involves comparing a given series with a known benchmark series. The core idea is that if a series with non-negative terms is less than or equal to a convergent series, then it also converges. Conversely, if it exceeds a divergent series, then it diverges.

Given the series: 4 + 1/5 + 0.3 + 1/3 + 2 + 1/9 + 3 + 1/27 + 4 + 1/81 + 5 + ..., we observe that the series potentially involves two components: a series of integers and a geometric series of reciprocals.

Noticing the pattern, the series combines a divergent part (the sum of integers 4 + 5 + ...) and a convergent geometric series (since the reciprocals decrease geometrically: 1/3, 1/9, 1/27, 1/81, ...). To evaluate its convergence, we focus on the geometric component, which converges by the geometric series test, because the common ratio is less than 1 in magnitude (e.g., 1/3, 1/9, etc.).

Comparing the divergent integer sum with the geometric component using the comparison test, because the sum of the integers diverges, the entire series diverges. Thus, the comparison test indicates divergence.

B. nth Term Test

The nth term test involves examining the limit of the general term of the series as n approaches infinity. If this limit does not approach zero, then the series diverges. If it approaches zero, the test is inconclusive.

For the series: ∑ (j² + 1)/(j²), the term is (j² + 1)/(j²) = 1 + 1/j².

Evaluating the limit as j approaches infinity: lim_j→∞ (1 + 1/j²) = 1 + 0 = 1.

Since this limit is not zero, the nth term test shows that the series diverges.

C. Integral Test

The integral test assesses the convergence of a series by integrating its corresponding continuous function. If the integral converges to a finite value, then the series converges; if not, it diverges.

Given the series: ∑ 1/(3k + 1) from k=1 to infinity, we consider the continuous function f(x) = 1/(3x + 1).

Evaluating the improper integral: ∫₁^∞ 1/(3x + 1) dx, we find:

Let u = 3x + 1, so du = 3 dx, thus dx = du/3.

The limits change from x=1 (u=4) to ∞ (u=∞):

∫_u=4^∞ 1/u * (du/3) = (1/3) ∫₄^∞ 1/u du = (1/3) [ln u]₄^∞ = (1/3) (lim_u→∞ ln u - ln 4).

Since ln u diverges to infinity as u approaches infinity, the integral diverges, thereby indicating that the series diverges by the integral test.

D. Alternating Series Test

The alternating series test states that if the absolute value of the terms decreases monotonically to zero, then the series converges.

Consider the series: ∑ (-1)^{i+1} (i + 3)/(i² + 10), from i=1 to infinity.

First, examine the absolute terms: a_i = (i + 3)/(i² + 10).

As i approaches infinity, a_i approaches 0 because the degree of the denominator exceeds the numerator, ensuring the terms tend to zero.

Next, verify monotonicity: The sequence a_i decreases for sufficiently large i, since numerator grows linearly and denominator quadratically.

Thus, by the alternating series test, the series converges.

E. Root Test

The root test involves taking the n-th root of the absolute value of the series' terms and evaluating the limit as n→∞. If the limit is less than 1, the series converges;

If greater than 1, it diverges; and if equal to 1, the test is inconclusive.

Given the series: ∑ 100 n e^{n}, for n=1 to infinity.

Calculate L = lim_n→∞ (|a_n|)^1/n = lim_n→∞ (100 n e^{n})^{1/n}.

Expressed as (100 n)^{1/n} (e^{n})^{1/n} = (100 n)^{1/n} e.

Now, (100 n)^{1/n} = e^{(1/n) ln(100 n)}. As n → ∞, (1/n) ln(100 n) → 0, hence (100 n)^{1/n} → e^{0} = 1.

Therefore, L = 1 * e = e, which is approximately 2.718 > 1.

The root test indicates divergence, since the limit exceeds 1.

Conclusion

The analysis demonstrates diverse behaviors in series convergence. The comparison and integral tests indicate divergence for the initial series. The nth term test confirms divergence for the series with terms approaching non-zero limits. The alternating series test confirms convergence under suitable conditions, and the root test corroborates divergence when the limit exceeds unity. Recognizing the applicability of each test allows mathematicians to accurately characterize infinite series.

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