Question

For each of the following sequences, if the divergence test applies, either state that \(\lim _{n \rightarrow \infty} a_{n}\) does not exist or find \(\lim _{n \rightarrow \infty} a_{n} .\) If the divergence test does not apply, state why. \(a_{n}=\frac{2^{n}}{3^{n / 2}}\)

Short answer

The sequence diverges as \( \lim_{n \to \infty} a_n = \infty \).

Step-by-step solution

Understand the Sequence

Consider the sequence given by \(a_{n} = \frac{2^{n}}{3^{n/2}}\). We need to determine whether this sequence converges or diverges using the divergence test.

Apply the Divergence Test

The divergence test states that if \( \lim_{n \to \infty} a_n eq 0\), then the series \( \sum a_n \) diverges. So first, we need to find \( \lim_{n \to \infty} a_n \).

Simplify the Sequence

Rearrange the expression to make it easier to find the limit:\[a_n = 2^{n} \cdot 3^{-n/2} = \left(\frac{2^2}{3}\right)^{n/2}\]

Evaluate the Limit

Evaluate \(\lim _{n \to \infty} a_{n} = \lim_{n \to \infty} \left(\frac{4}{3}\right)^{n/2}\). Since \( \frac{4}{3} > 1 \), as \( n \to \infty \), \( \left(\frac{4}{3}\right)^{n/2} \to \infty \).

Conclusion of the Divergence Test

Since \( \lim_{n \to \infty} a_n = \infty \), it is not equal to zero, and thus by the divergence test, the series \( \sum a_n \) diverges.

Key concepts

Divergence Test

The Divergence Test is an important tool for identifying the behavior of a series, especially when it comes to understanding whether it converges or diverges. When you have a sequence, you're trying to figure out what happens as you keep going further and further out. Here's how the Divergence Test can help:

• If the limit of a sequence as it approaches infinity is not zero, or it doesn't exist, then the sequence definitely doesn't converge.
• The keyword to remember is 'not zero.' If it stays away from zero, the sequence spreads out more and doesn't settle down.

It's important to note, however, that the Divergence Test itself cannot conclusively determine convergence. It only establishes divergence if the limit is non-zero or does not exist. If the limit is zero, further tests are needed to check for convergence.

Limit Evaluation

Evaluating limits is at the core of determining how sequences behave as they approach infinity. In this case, the sequence given by\(a_{n} = \frac{2^{n}}{3^{n/2}}\) requires manipulation to evaluate its limit.To simplify it, rewrite the expression like this:\[a_n = 2^{n} \cdot 3^{-n/2} = \left(\frac{4}{3}\right)^{n/2}\]This step makes it clear what happens as \(n\) grows very large:

• \(\frac{4}{3}\) is a number larger than one.
• Raising it to higher said yet larger powers incrementally pushes it further toward infinity.

The key takeaway is if the base of an exponential expression is greater than one, and the power grows unbounded, the outcome is that the expression will diverge to infinity.

Exponential Growth

Exponential growth refers to a process where the rate of change of a quantity is proportional to the current amount of the quantity itself. In mathematics, this typically involves expressions such as: \(b^n\) where \(b\) is a base greater than one.For our sequence, we saw: \[\left(\frac{4}{3}\right)^{n/2}\]This expression exhibits exponential growth due to the base \(\frac{4}{3}\) being greater than one:

• As \(n\) increases, each multiplication scales the value up rapidly.
• This growth pace is much faster compared to linear growth or even polynomial growth, making exponential growth challenging to contain as it eventually heads toward infinity.

In practical sense, exponential growth models depict rapid and accelerating increase, found often in fields such as population dynamics, finance, and computer science.