Question

Use the Ratio Test to determine the convergence or divergence of the series. $$ \sum_{n=0}^{\infty} \frac{3^{n}}{(n+1)^{n}} $$

Short answer

The series diverges according to the Ratio Test.

Step-by-step solution

Setup the Ratio Test

First set up the limit one would like to find as per the Ratio Test, \( \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| \), where \( a_n = \frac{3^{n}}{(n+1)^{n}} \).

Analyze the ratio

Next, plug in \( n+1 \) wherever n is in the nth term to find \( a_{n+1} \). This gives \( a_{n+1} = \frac{3^{n+1}}{(n+2)^{n+1}} \). The ratio hence becomes \( \frac{3^{n+1}}{(n+2)^{n+1}} \div \frac{3^{n}}{(n+1)^{n}} \). This simplifies to \( \frac{3^{n+1}(n+1)^{n}}{3^{n}(n+2)^{n+1}} \)

Simplify the Ratio

The powers of 3 and terms in the denominator simplify to: \( \frac{3(n+1)^{n}}{(n+2)^{n}} \). Now \( \lim_{n \to \infty} \frac{3(n+1)^{n}}{(n+2)^{n}} \) needs to be determined.

Find the Limit

Rewrite the limit as the product of two separate limits: \( \lim_{n \to \infty} \frac{(n+1)^{n}}{(n+2)^{n}} \times \lim_{n \to \infty} 3 \). We need to find the first limit, as the limit of 3 is simply 3. By using l'Hopital's rule one gets \( \lim_{n \to \infty} \frac{(n+1)^{n}}{(n+2)^{n}} = \lim_{n \to \infty} \frac{n \cdot (\ln(n+1))^{n-1}}{(n+2) \cdot (\ln(n+2))^{n+1}} = 1 \). The ratio of the n+1 and nth terms is hence 3.

Convergence or Divergence

Since the limit of the ratio of the \( n+1 \)th term to the nth term is greater than 1, the given series diverges by the Ratio Test.

Key concepts

Series Convergence

Understanding when a series converges is a crucial aspect of studying infinite series in mathematics. A series is essentially a sum of the terms of a sequence. Convergence of a series means that as you sum more and more terms, the total approaches a specific value, called the sum of the series. In contrast, if the series does not approach any particular value as more terms are added, we say the series diverges.

One effective tool to determine series convergence is the Ratio Test, which involves looking at the limit of the absolute value of the ratio of consecutive terms. Specifically, for a series given by the sum of terms a_n, the Ratio Test directs us to find lim_n→∞ |a_n+1/a_n|. If this limit is less than 1, the series converges. If it is greater than 1, the series diverges, and if it is equal to 1, the test is inconclusive and additional methods need to be used to determine convergence.

In the case of the series ∑_n=0^∞ 3ⁿ/(n+1)ⁿ, using the Ratio Test indicates that the series diverges since the limit is greater than 1. It is important when applying this test to ensure that the limit is calculated correctly, which often necessitates understanding of additional concepts such as the limit of a sequence.

Limit of a Sequence

The limit of a sequence is a fundamental concept in calculus that describes the behavior of a sequence as its index, usually denoted as n, approaches infinity. If the sequence approaches a particular value, that value is called the limit of the sequence.

Formally, a sequence a_n has a limit L if for any positive number ε (no matter how small), there exists a corresponding positive integer N such that for all n > N, the absolute difference |a_n - L| is less than ε. This notion can be intuitively understood as the idea that the terms of the sequence get arbitrarily close to L as n becomes large.

In practice, finding the limit of a sequence can sometimes be straightforward, but in other cases, it requires more advanced methods like l'Hopital's rule or special techniques for handling indeterminate forms. In the example series given earlier, finding the limit of the ratio of consecutive terms requires careful manipulation and understanding of limits.

L'Hopital's Rule

In the world of calculus, l'Hopital's rule is a method for finding limits that lends a hand when more straightforward approaches are ineffective. This rule is applicable in situations where calculating the limit leads to an indeterminate form such as 0/0 or ∞/∞. L'Hopital's rule states that if the limits of functions f(x) and g(x) both approach 0 or both approach ∞ as x approaches c, and the derivatives f'(x) and g'(x) are continuous near c, then lim_x→c (f(x)/g(x)) = lim_x→c (f'(x)/g'(x)), provided that the limit on the right side exists or is ∞.

An example of using l'Hopital's rule can be found in the provided textbook solution where the limit lim_n→∞ ((n+1)ⁿ/(n+2)ⁿ) is calculated. Since direct substitution of infinity would result in an indeterminate form, the rule is employed, and subsequent differentiation leads us to understand that the given limit equals 1. It is a powerful technique that often simplifies complex limit problems, however, it is important to make sure that its conditions are met before applying it.