How to Spot Divergent Series Using Simple Tests

Math can be pretty cool, especially when we talk about sequences and series. One important idea we come across is whether a series converges or diverges.

This means we want to know if a series reaches a certain limit (convergence) or if it keeps going forever (divergence). Let’s look at some easy tests to tell if a series diverges.

What is a Series?

First, let’s break down what we mean by a series.

• A finite series is when you add a set number of terms together. Think about this example: $1 + 2 + 3 + 4 + 5 = 15$

• An infinite series keeps going forever. For instance: $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots$ (This series actually converges.)

How to Identify Divergent Series

So, how do we know if an infinite series diverges? Here are some simple tests you can use:

1. The Divergence Test: This is one of the easiest methods. If the series terms don’t get closer to zero, then the series diverges.

   Example: $1 + 1 + 1 + 1 + \ldots$ Here, the terms are always $1$. Since they don’t get closer to $0$, this series diverges.

2. The p-Series Test: A p-series looks like this: $\sum_{n=1}^{\infty} \frac{1}{n^p}$ For divergence, if $p$ is less than or equal to 1, then the series diverges.

   • For example, if $p = 1$: $\sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + ...$ This is called the harmonic series, and it diverges.
   • But for $p = 2$: $\sum_{n=1}^{\infty} \frac{1}{n^2}$ This one converges.

3. The Comparison Test: You can compare your series to a known divergent series. If your series is larger than the divergent one after a certain point, then your series also diverges.

   Example: Compare: $\sum_{n=1}^{\infty} \frac{1}{n}$ (which diverges) with: $\sum_{n=1}^{\infty} \frac{2}{n}$ Since $\frac{2}{n}$ is always bigger than $\frac{1}{n}$ for all $n \geq 1$, the comparison test shows that this series diverges too.

4. Ratio Test: This test helps for series that involve factorials or exponential numbers. If we find this limit: $L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$ and it’s greater than $1$, then the series diverges. If it’s less than $1$, it converges.

   Example: Consider the series: $\sum_{n=1}^{\infty} n!$ Using the ratio test, we can see that it diverges since the terms grow really fast.

Conclusion

In short, figuring out if series diverge is not as hard as it seems. You can use simple tests like the divergence test, p-series test, comparison test, and ratio test. These tools help you understand how infinite series behave and improve your knowledge of sequences and series in pre-calculus. So, when you bump into an infinite series next time, remember these tests and check if it converges or diverges! Happy studying!