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. $$\sum_{n=1}^{\infty} \frac{n+3}{n^{2}+10 n}$$

Short answer

The preliminary test is inconclusive because the limit of the general term is zero.

Step-by-step solution

Examine the Series

The series to analyze is \ \(\sum_{n=1}^{\text{\infty}} \frac{n+3}{n^{2}+10n} \).

Simplify the General Term

We start by simplifying the general term\: \ \( \frac{n+3}{n^{2}+10n} = \frac{n+3}{n(n+10)}\).

Use the Preliminary Test

Check the limit of the general term as \( n \) approaches infinity.\ \[\lim_{{n \to \infty}} \frac{n+3}{n(n+10)} = \lim_{{n \to \infty}} \frac{1 + \frac{3}{n}}{n + 10}\].

Evaluate the Limit

Evaluate the limit by considering the behavior of each term as \( n \) approaches infinity:\ \[\lim_{{n \to \infty}} \frac{1 + \frac{3}{n}}{n + 10} = \frac{1+0}{\infty+10} = 0\].

Apply the Conclusion of the Preliminary Test

Since the limit of the general term as \( n \) approaches infinity is zero, the preliminary test is inconclusive. Formally, the preliminary test does not confirm divergence.

Key concepts

Convergence Criteria

When working with infinite series, one key concept is determining whether a series converges or diverges. Convergence implies that as you add infinite terms, the sum approaches a finite value. Divergence means the sum grows without bound or oscillates indefinitely.
In this exercise, we used the Divergence Test (also known as the Preliminary Test) to evaluate the series \(\sum_\{n=1\}\^{\infty} \frac{n+3}{n^{2}+10n}\)\. This test checks if the limit of the general term as \( n \) approaches infinity is zero. If it does not, the series diverges. Conversely, if the limit is zero, it indicates nothing conclusive about convergence; further tests are required.
For any series \( \sum_{n=1}^{\text{\textbackslash infty}} a_{n}\ \),\ we calculate \(\text{\textbackslash lim}_{n\{\text{\textbackslash to}\textbackslash infty}} a_{n} \). If the limit \( \text{\textbackslash lim}_{n\{\text{\textbackslash to}\textbackslash infty}} a_{n} != 0 \), the series diverges.

Series Simplification

Simplifying the general term of a series is crucial for applying tests like the Divergence Test. Simplification helps in understanding the behavior of the general term as the variable approaches infinity.
In this problem, we simplified \(\frac{n+3}{n^{2}+10n} \) to \(\frac{n+3}{n(n+10)}\)\. Such simplifications make it easier to identify dominant terms and apply limits effectively.
Remember, isolating the largest power of \( n \) in the numerator and denominator helps reveal the term's behavior as \( n \) becomes very large. In our case, \(\frac{n+3}{n(n+10)}\)\ allowed us to express the general term more clearly for evaluation.

Limit Evaluation

Limit evaluation is a mathematical approach to study how a function behaves as the input grows indefinitely. For series analysis, it helps determine the long-term behavior of the terms.
To evaluate \(\lim_{{n \to \infty}} \frac{n+3}{n(n+10)}\)\, we simplify and split terms to handle limits more efficiently. We rewrote it as: \(\lim_{{n \to \infty}} \frac{1+\{\frac{3}{n}}}{n+10}\)\. Evaluating as \( n \to \infty \) gives \(\frac{1+0}{\textbackslash infty+10} = 0\)\.
This limit being zero told us the Divergence Test was inconclusive for this series, leading us to the conclusion we needed to employ additional convergence tests.