Problem 52 - Thomas Calculus

Question

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{3^{n}}{n^{3}} $$

Step-by-Step Solution

Answer

The sequence \(a_n\) diverges as \(n \to \infty\).

Step 1: Determine the Limit Behavior

Examine the form of the sequence \( a_n = \frac{3^n}{n^3} \). The numerator \( 3^n \) grows exponentially while the denominator \( n^3 \) grows polynomially. As \( n \to \infty \), the exponential growth of \( 3^n \) will dominate the polynomial growth of \( n^3 \). Therefore, we expect the sequence \( a_n \) to diverge.

Step 2: Use the Ratio Test

To confirm divergence, apply the ratio test by considering \( \frac{a_{n+1}}{a_n} \). Compute:\[\frac{a_{n+1}}{a_n} = \frac{\frac{3^{n+1}}{(n+1)^3}}{\frac{3^n}{n^3}} = \frac{3 \cdot 3^n \cdot n^3}{3^n \cdot (n+1)^3} = \frac{3n^3}{(n+1)^3}.\]As \( n \to \infty \), this ratio approaches \( 3 \). Since this limit is greater than 1, by the ratio test, the sequence diverges.

Key Concepts

Limit Behavior

When determining if a sequence converges or diverges, analyzing its limit behavior is crucial. This means understanding what happens to the sequence's terms as they extend towards infinity (i.e., as \( n \to \infty \)). In the given example, the sequence is \( a_n = \frac{3^n}{n^3} \). The numerator, \( 3^n \), shows exponential growth, which occurs at a much faster rate compared to polynomial growth exhibited by the denominator, \( n^3 \). As \( n \) becomes very large, the impact of \( 3^n \) escalates significantly more than \( n^3 \). This suggests that the sequence does not stabilize at any finite value, indicating divergence. The concept of limit behavior here helps us foresee that the sequence will not converge to a specific number but instead head towards infinity or away from zero.

Ratio Test

The ratio test is a handy tool for examining the convergence or divergence of a sequence or series. It involves comparing successive terms of the series, specifically by taking the limit of \( \frac{a_{n+1}}{a_n} \) as \( n \to \infty \). In the sequence \( a_n = \frac{3^n}{n^3} \), the ratio test illuminates why the sequence diverges. Calculating \( \frac{a_{n+1}}{a_n} \), we determine:

• \( \frac{a_{n+1}}{a_n} = \frac{\frac{3^{n+1}}{(n+1)^3}}{\frac{3^n}{n^3}} = \frac{3n^3}{(n+1)^3} \)
• As \( n \to \infty \), this simplifies to \( 3 \)

Since the ratio is greater than 1, the series diverges, as per the ratio test. The ratio test is particularly effective in sequences where terms involve exponential and polynomial components, as it provides a concrete threshold for convergence.

Exponential Growth

Understanding exponential growth is key when analyzing sequences like \( a_n = \frac{3^n}{n^3} \). Exponential growth means a quantity increases at a rate proportional to its current value. In mathematical terms, if \( a \) is a constant greater than 1, \( a^n \) (such as \( 3^n \) here) will grow exponentially. This type of growth is incredibly rapid. As \( n \) increases, the quantity \( 3^n \) becomes extremely large, much faster than any polynomial growth counterpart. This is why, in the sequence provided, \( 3^n \) overshadows \( n^3 \) as \( n \to \infty \), leading the whole sequence to diverge. Thus, the behavior of exponential growth is so dominant that it determines the overall sequence behavior.

Polynomial Growth

In contrast to exponential growth, polynomial growth refers to an increase at a fixed power rate. For example, \( n^3 \) shows polynomial growth of degree 3. This type of growth is steady and predictable, increasing as a power of \( n \). While still robust, polynomial growth cannot match the velocity of exponential growth as \( n \to \infty \). In the sequence \( a_n = \frac{3^n}{n^3} \), the denominator \( n^3 \) grows polynomially. While it becomes large with increasing \( n \), it does so at a slower pace than the numerator \( 3^n \). Recognizing this growth distinction is vital. It helps explain why sequences with exponentially growing numerators and polynomially growing denominators, like this one, tend to diverge due to the overpowering speed of exponential growth over polynomial.