Sequence Rules: Convergence
What does it feel like to invent math?
We've found that some series have closed forms for an infinite number of terms, and some don't. If an infinite series has one, non-infinite solution, it is called convergent. If it does not, it is called divergent.
There are some rules and tests we can use to test if a series is convergent. These are beyond the scope of the course, but curious students may research this topic to see some interesting examples of convergent series.
The series $1 - 1 + 1 - 1 + 1 + ...$ diverges, since it cannot be determined whether the value will be $1$ or $0$. The series $1 + 2 + 3 + 4 + ...$ and $1 + 1 + 1 + 1 + ...$ also diverge since they are infinite. The series $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + ...$ is called the Harmonic series, and is known to be divergent.
However, $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - ...$ converges to a strange value, $\ln{(2)}$.
It can be very hard to tell which series are convergent or divergent, but it is important to consider this before completing any math with series, as ignoring the convergence and divergence of series can have some interesting results.
Breaking Sequence Rules
What does it feel like to invent math?
To come up with values that "make sense" in the normal idea of mathematics, we must stick to the rules of series. However, when we don't follow these rules, some interesting math appears. Watch the following 3Blue1Brown video on the topic. You may have to pause or rewind the video to think about what the narrator is saying.
This math is not the typical math most people encounter, however, it is a valid field with some interesting applications. Often, even the math that makes no sense at first becomes applicable to real life scenarios.
In our exercises, we will stick to simpler series, but the ideas behind this different math are interesting enough to discuss.