Question

Use the preliminary test to decide whether the following series are divergent or require further testing. Careful: Do not say that a series is convergent; the preliminary test cannot decide this. $$\sum_{n=1}^{\infty} \frac{3^{n}}{2^{n}+3^{n}}$$

Short answer

The series \( \sum_{{n=1}}^{{\ to\ \infty}} \frac{3^{n}}{2^{n}+3^{n}} \) diverges.

Step-by-step solution

Understand the Preliminary Test

The preliminary test, or divergence test, states that if \( \lim_{{n \to \infty}} a_n \eq 0\) or does not exist, then the series \( \sum a_n \) diverges.

Identify the General Term

The general term for the given series is \( a_n = \frac{3^{n}}{2^{n}+3^{n}} \).

Compute the Limit of the General Term

To use the preliminary test, compute \( \lim_{{n \to \infty}} \frac{3^{n}}{2^{n}+3^{n}} \). First divide the numerator and the denominator by \( 3^{n} \) to simplify: \[ a_n = \frac{3^{n}}{2^{n}+3^{n}} = \frac{3^{n}/3^{n}}{2^{n}/3^{n} + 3^{n}/3^{n}} = \frac{1}{(2/3)^{n} + 1} \] As \ to\ \infty \, \ (2/3)^n\ to\ 0 \. Therefore, \ \lim_{{n \to \infty}} \frac{1}{(2/3)^{n} + 1} = 1 \.

Apply the Preliminary Test

Since the limit \( \lim_{{n \to \infty}} \frac{3^{n}}{2^{n}+3^{n}} = 1 \) is not equal to 0, the preliminary test indicates that the series \( \sum_{{n=1}}^{{\infty}} \frac{3^{n}}{2^{n}+3^{n}} \) diverges.

Key concepts

Understanding Infinite Series

In mathematics, an infinite series is a sum of infinitely many terms. Instead of stopping after a finite number of terms, you keep adding more and more.
This doesn't mean the series goes on forever without adding up to something meaningful.
Some infinite series can add up to specific values, a concept known as convergence.
Others don't add up neatly and are called divergent.
When working with infinite series, we are often interested in their behavior as the number of terms grows larger and larger.
Think of an infinite series like the sum of an ever-growing list.
For instance, imagine rotating through an infinite series of numbers 1, 1/2, 1/3, ... endlessly.
This series is known as the Harmonic Series, and while the individual terms get smaller, the sum keeps growing and does not settle at a particular value.
This Harmonic Series is an example of a divergent series. To deal with series, mathematicians use tests, like the preliminary (divergence) test, to identify which series diverge (don't settle) and which might converge (settle).

Using the Preliminary Test for Divergence

The preliminary test, also known as the divergence test, is a quick way to check if a series diverges.
The test involves looking at the limit of the general term in the series as the index goes to infinity.
Here's how it works:

• If the limit of the general term is not zero or does not exist, the series diverges.
• If the limit is zero, the test is inconclusive, and we need further testing to see if the series converges.

To apply this to our example series, let's analyze its general term \( a_n = \frac{3^{n}}{2^{n}+3^{n}} \).
We simplify this term by dividing both the numerator and the denominator by \( 3^n \).
This gives us \( \frac{1}{(2/3)^n + 1} \).
Now, observe that as \( n \) goes to infinity, \( (2/3)^n \) approaches 0.
So, the term simplifies to \( \frac{1}{1 + 0} = 1 \).
This limit, being 1 and not 0, tells us that the series diverges.

The Role of Limits in Series Tests

The limit of a sequence plays a crucial role in determining the behavior of an infinite series.
In calculus, calculating limits helps us understand where a sequence is headed as its index becomes very large.
For a sequence \( a_n \), if \( \text{lim}_{n \to fty} a_n = L \), we are saying that the terms of the sequence get closer to \( L \) as \( n \) increases.
This concept is essential for the preliminary test.
Taking our example again, the sequence term is \( a_n = \rray{3^{n}}{2^{n}+3^{n}} \).
By simplifying, we see that the terms tend towards 1 as \( n \) grows.
Hence, \( \text{lim}_{n \to fty} a_n eq 0 \), which indicates that the series diverges.
Remember, even if the limit is zero, it doesn't guarantee convergence.
It's only one of many conditions to check if an infinite series converges.
Thus, the limit provides a starting point that often determines whether further tests are needed.