Question

Determine the convergence of the given series using the Root Test. If the Root Test is inconclusive, state so and determine convergence with another test. $$\sum_{n=1}^{\infty} \frac{4^{n+7}}{7^{n}}$$

Short answer

The series converges by the Root Test.

Step-by-step solution

Identify the series and terms

The series given is \( \sum_{n=1}^{\infty} \frac{4^{n+7}}{7^n} \). Let the general term be \( a_n = \frac{4^{n+7}}{7^n} \). We need to apply the Root Test to this series.

Apply the Root Test

According to the Root Test, we need to compute the \( n \)-th root of the absolute value of the term \( a_n \). Specifically, we calculate \( \lim_{n \to \infty} \sqrt[n]{|a_n|} \). Here, \( |a_n| = \frac{4^{n+7}}{7^n} = \frac{4^7 \cdot 4^n}{7^n} = 4^7 \cdot \left(\frac{4}{7}\right)^n \).

Simplify the expression

Calculate the \( n \)-th root: \[ \lim_{n \to \infty} \sqrt[n]{4^7 \cdot \left(\frac{4}{7}\right)^n} = \lim_{n \to \infty} \left(4^7\right)^{1/n} \cdot \left(\frac{4}{7}\right) = 1 \cdot \frac{4}{7} = \frac{4}{7} \]The expression simplifies to \( \frac{4}{7} \).

Analyze the Root Test Condition

For the series to converge, the limit found in the Root Test, \( \frac{4}{7} \), must be less than 1. Since \( \frac{4}{7} = 0.571 < 1 \), the Root Test indicates that the series converges.

Conclusion and Confirmation

The Root Test indicates convergence since the limit is less than 1. Therefore, the series \( \sum_{n=1}^{\infty} \frac{4^{n+7}}{7^n} \) converges. There is no need for another test as the Root Test was conclusive.

Key concepts

Root Test

The Root Test is a valuable tool in determining whether an infinite series converges or diverges. It is especially useful for series expressed in terms that are raised to the power of the index. Let's break down the application of the Root Test in simpler terms.

To use the Root Test, you first take the general term of the series, say \( a_n \). You then compute the \( n \)-th root of its absolute value, \( \sqrt[n]{|a_n|} \). The next step is to take the limit as \( n \) approaches infinity: \( \lim_{n \to \infty} \sqrt[n]{|a_n|} \).

Here's how to interpret the result:

• If the limit is less than 1, the series converges absolutely.
• If the limit is greater than 1, the series diverges.
• If the limit equals 1, the test is inconclusive, and another method must be used.

This makes the Root Test particularly neat and tidy compared to some other tests, which is why it's often a first choice for series involving exponentiation. Remember, practice makes perfect, and applying the Root Test multiple times will help you understand its nuances.

Infinite Series

An infinite series is simply a sum that continues indefinitely. It involves adding an infinite number of terms. These terms form what's called a sequence. Think of it like an endless summation where you keep adding terms forever.

There are plenty of infinite series in mathematics, each with unique characteristics. Some common examples include:

• Geometric series, where each term is a constant multiple of the previous one.
• Arithmetic series, though technically these are finite, have their infinite counterparts in concepts such as arithmetic progressions.
• Harmonic series, where terms are the reciprocals of natural numbers.

Understanding infinite series is crucial because they provide the framework for numerous applications in math, physics, and engineering, among others. When dealing with infinite series, one essential question is whether the series converges, meaning it approaches a specific value, or diverges, meaning it grows without bound.

Convergence Tests

Convergence tests are methods that help us determine whether an infinite series converges or diverges. There are several tests available, each more suited to different kinds of series. Let's discuss a few commonly used ones.

**Root Test:** As described earlier, this test is particularly useful for series involving powers, aiding in quickly identifying convergence or divergence.

**Ratio Test:** Similar to the Root Test but instead uses the ratio of consecutive terms to find convergence. It's especially handy for factorials or exponential functions.

**Integral Test:** Relates a series to an integral. If the integral converges, so does the series, and vice versa.

Each convergence test has its scope of application, helping us understand the behavior of a series. The key is to pick the appropriate test for the series at hand. With practice, recognizing which test to use for given circumstances becomes intuitive. Thus, convergence tests are like a toolbox where each test is a different tool to solve the puzzle of series convergence.