The Limit Comparison Test: A Simple Guide

When studying series and whether they converge (come together) or diverge (move apart), the Limit Comparison Test is an important tool. It helps us look at two series to see if they behave the same way.

You should use the Limit Comparison Test when you have a series with positive terms. This is especially useful if it's hard to tell if a series converges just by looking at it.

Using the Limit Comparison Test

To use this test, you first pick a comparison series. This could be a well-known series, like a $p$-series or a geometric series. These series have clear rules about whether they converge or diverge.

For example, let’s say you have a series called $\sum a_n$. You will compare it with another series called $\sum b_n$, which is easier to work with because its behavior is known. Ideally, the two series should act similarly when you look at larger values of $n$.

Steps for the Limit Comparison Test

1. Positive Terms: Make sure both series only contain positive terms. This is important because using negative or mixed terms can make the test confusing.

2. Find the Limit: You need to calculate the following limit:

   $L = \lim_{n \to \infty} \frac{a_n}{b_n}.$

3. Check the Value of L: Now, look at the value of $L$:

   • If $0 < L < \infty$, this tells you both series $\sum a_n$ and $\sum b_n$ either both converge or both diverge.
   • If $L = 0$, then $\sum a_n$ converges if $\sum b_n$ converges. If $\sum b_n$ diverges, then $\sum a_n$ does too.
   • If $L = \infty$, it means $\sum a_n$ diverges if $\sum b_n$ diverges.

Examples of Simple Series for Comparison

• $p$-Series: This is written as $\sum \frac{1}{n^p}$. It converges when $p > 1$ and diverges when $p \leq 1$.
• Geometric Series: This has the form $\sum ar^n$. It converges if the common ratio $|r| < 1$ and diverges if $|r| \geq 1$.

When to Use the Limit Comparison Test

You might want to use the Limit Comparison Test when:

• The terms in your series $a_n$ are complicated or don't easily compare to simpler series.
• You think your series looks like a $p$-series or a geometric series, but you're not sure without doing some calculations.

Conclusion

In summary, the Limit Comparison Test is a helpful method to determine whether series converge or diverge, especially in Calculus II classes. It helps you compare hard-to-understand series with those that are easier to analyze. So, the next time you have a series of positive terms that seem tricky, remember to consider the Limit Comparison Test. It can make finding out about convergence much simpler!