Question

Use the Direct Comparison Test to determine the convergence of the given series; state what series is used for comparison. $$\sum_{n=1}^{\infty} \frac{2^{n}}{5^{n}+10}$$

Short answer

The series converges by comparing with the convergent geometric series \( \sum_{n=1}^{\infty} \left( \frac{2}{5} \right)^n \).

Step-by-step solution

Understanding the Given Series

We are given the series \( \sum_{n=1}^{\infty} \frac{2^{n}}{5^{n}+10} \). This is our point of focus for determining convergence using the Direct Comparison Test. The first task is to identify the general term of this series, which is \( a_n = \frac{2^n}{5^n + 10} \).

Choose a Series for Comparison

For the Direct Comparison Test, we need a known series to compare against. Noting that \( 5^n + 10 \) is greater than \( 5^n \), we observe that \( \frac{2^n}{5^n + 10} < \frac{2^n}{5^n} = \left( \frac{2}{5} \right)^n \). The series \( \sum_{n=1}^{\infty} \left( \frac{2}{5} \right)^n \) is a geometric series.

Evaluate the Comparison Series

The geometric series \( \sum_{n=1}^{\infty} \left( \frac{2}{5} \right)^n \) is convergent because the common ratio, \( \frac{2}{5} \), is less than 1. This information is key since a convergent series with larger terms can help demonstrate the convergence of our series using the Comparison Test.

Apply the Direct Comparison Test

Since \( \frac{2^n}{5^n + 10} < \left( \frac{2}{5} \right)^n \) and \( \sum_{n=1}^{\infty} \left( \frac{2}{5} \right)^n \) converges, by the Direct Comparison Test, the series \( \sum_{n=1}^{\infty} \frac{2^{n}}{5^{n}+10} \) also converges.

Key concepts

Understanding Geometric Series

A geometric series is one of the most fundamental concepts in mathematics, particularly useful in the study of series and sequences. It is a series of the form \( \sum_{n=0}^{\infty} ar^n \), where \( a \) is the first term and \( r \) is the common ratio. Each term is the product of the previous term and the common ratio. If you know the first term and the common ratio, you can determine any term in the series.

For geometric series, convergence depends on the common ratio \( r \):

• If \( |r| < 1 \), the series converges.
• If \( |r| \geq 1 \), the series diverges.

Convergence here means that as you add more and more terms, the sum approaches a specific finite value. This characteristic makes geometric series a handy tool in solving problems involving infinite series, as they provide a clear criterion for convergence based on only one parameter, the common ratio.

Exploring the Convergence of Series

Convergence is a vital property when dealing with series in mathematical analysis. It tells us whether a series will add up to a finite limit, no matter how many terms you add.

A series \( \sum_{n=1}^{\infty} a_n \) is said to be convergent if the sequence of its partial sums \( S_n = a_1 + a_2 + \ldots + a_n \) approaches a finite limit as \( n \to \infty \).

• If the limit of \( S_n \) exists and is finite, the series converges.
• If the limit does not exist or is infinite, the series diverges.

Understanding the convergence of a series is crucial because it dictates the behavior of the series over an infinite number of terms. It allows mathematicians and scientists to predict and model real-world phenomena accurately. Techniques like the Direct Comparison Test are used to determine the convergence by comparing with a known convergent or divergent series.

Utilizing Comparison Series in the Direct Comparison Test

The Direct Comparison Test is a method used to determine if a given series converges by comparing it with another series whose convergence properties are well-known. It's like using a benchmark series to evaluate another's behavior.

The basic idea involves:

• Selecting a comparison series \( \sum b_n \) such that for all sufficiently large \( n \), \( 0 \leq a_n \leq b_n \).
• If \( \sum b_n \) converges and \( a_n \leq b_n \), then \( \sum a_n \) also converges.
• If \( \sum b_n \) diverges and \( a_n \geq b_n \), then \( \sum a_n \) also diverges.

The choice of comparison series is critical. In practice, good candidates are often geometric series due to their straightforward criterion for convergence. By demonstrating the convergence of the comparison series, particularly when dealing with terms that simplify to a geometric nature, one can conclude the convergence of the series under evaluation.