Learn how mathematical series and the compound interest formula can be used in the real world to calculate how fast money grows in recurring investments.

In the past several articles, we talked about compound interest and its relevance your savings account. We also discussed a much more effective way of saving money called a recurring investment. Today, we’re going to finish up this series with a look at how mathematical series can help you quickly calculate how much a recurring investment—such as a retirement account—will be worth next year, next decade, or even way out in the distant future.

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## Recap: What are Mathematical Series?

Before we dive into the relationship between mathematical series and recurring investments, let’s take a minute to remind ourselves about the meaning of a series in math. The idea is pretty simple: A series is just the sum of the numbers in a sequence (which is basically just a list of numbers). Some contain a non-infinite number of terms (those are just the parts of a series that you add up) and are therefore called finite series. For example, the series you get by adding up all the squares of the integers between 1 and 10 is a finite series since it has only 10 terms. Other series contain an infinite number of terms and are therefore called infinite series. For example, the series we got by dividing up a square an infinite number of times a few articles ago was an infinite series.

## What Does “Infinite” Mean in an Infinite Series?

Which brings us to a great question that math fan, Ian, asked in an email. Ian pointed out that adding up the infinite number of integers on the number line must always equal zero. In other words, since every positive integer has a corresponding negative version of itself on the other side of the number line, the total sum of all the integers must be zero. Which, Ian asks, means that all infinite series must add to zero too, right? If that’s the case, what’s the point of an infinite series?

Ian’s question brings up a really interesting topic: Namely, that there’s more than one type of infinity. We’ll talk more about this in the future, but for now just know that while it’s true that adding up the infinite number of integers gives zero, not every infinite series has to include all the integers. For example, there is an infinite number of even numbers, an infinite number of odd numbers, an infinite number of numbers that are evenly divisible by 17, and an infinite number of other possibilities. The key point is that any of these other infinite sets of numbers can be used to create infinite series that don’t necessarily add to zero. So, to answer Ian’s question: No, not all infinite series add up to zero because the answer depends on exactly what infinite group of numbers we’re talking about adding up.