How to Use Mathematical Series in the Real World
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.
Jason Marshall, PhD
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How to Use Mathematical Series in the Real World
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.
How Does Money Grow in Recurring Investments?
Now that we all remember what mathematical series are, we’re ready to see how they apply to the real world. In this particular case, the series we’re talking about is a finite series, and it’s related to the idea of a recurring investment that we talked about in the last article. As you’ll recall, a recurring investment is one in which money is periodically put into an interest-earning account.
To see how this works, let’s review the basic way to calculate the value of such an investment some number of years after you start saving your money. In particular, let’s say that you invest $2000 at the start of every year into an account earning 5% interest. Using the compound interest formula, we find that the value of the account at the end of the first 3 years is:
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Value at End of Year 1 = $2000 x 1.05^1 = $2100
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Value at End of Year 2 = $2000 x 1.05^1 + $2000 x 1.05^2 = $4305
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Value at End of Year 3 = $2000 x 1.05^1 + $2000 x 1.05^2 + $2000 x 1.05^3 = $6620.25
Are you noticing a pattern? If you look closely, you’ll see that this sequence of expressions for the value of the investment at the end of each year form a mathematical series where each term is the amount of money invested each year—$2000—multiplied by the interest rate raised to a higher and higher integer power.
How Mathematical Series Solve Real World Problems
As it turns out, we can use this fact to create a formula analogous to the compound interest formula that allows us to find the value of the account any number of years in the future. Why is that such a big deal? Because it lets us skip all the work of calculating the value at the end of each year, and instead jump right to the final value in the future. We won’t go through every last detail right now (because it gets a little complicated), but if you’re curious to see those details they’re available as a bonus section at the end of this article.
[[AdMiddle]For now, all you need to know is that since the expressions for the value of a recurring investment form a mathematical series, it’s possible to use some cool (but slightly more advanced) math to figure out what that series is going to add up to. The end result is a formula for the value of a recurring investment at any point in the future. The formula says:
(future value) = (previous value) x [ (1+rate) / rate ] x [ (1+rate)^years – 1 ]
Yes, I know that’s a mouthful, but try it out and you’ll see that it really does work for 3, 5, 20, 100, or whatever number of years you want! It’s a very handy formula to have in your bag of mathematical tricks.
Number of the Week
Before we finish up, it’s time for this week’s featured number selected from the various numbers of the day posted to the Math Dude’s Facebook page. This week’s number is 10^22…that is, a 1 with 22 zeros after it. What could a number that big possibly be? Well, it’s an estimate of the number of grains of sand on all the beaches of the world. While that’s a huge number, what’s even more incredible is that the new system for assigning addresses to Internet devices that was tested last week contains enough addresses to assign one to each of those 10^22 grains of sand on Earth…300 trillion times! So, I think we’re all set for Internet addresses for a while.
For the bonus tip, click Click here for more information.
New FREE Math Dude E-book!
Have you heard the news? My new e-book, Why Math Isn’t an Awful Nerd, is available for download for FREE right now from Amazon, Barnes & Noble, and the iBookstore. What’s it about? Well, the title really says it all: A lot of people have had bad experiences with math, and they feel like it’s just a painful and awful subject. I understand where they’re coming from, but I also know that it really doesn’t have to be that way. So, using lots of visual explanations, puzzles, and games, this new ebook explains step-by-step exactly how math works, why it’s so awesome, and hopefully it’ll convince all those people thinking that math is awful to change their minds and agree with me that math is—dare I say it—elegant and even beautiful. Be sure to download your copy of Why Math Isn’t an Awful Nerd today! Thanks!
Wrap Up
Okay, that’s all for today. Remember to become a fan of the Math Dude on Facebook where you’ll find a new number of the day and math puzzle posted each and every weekday. And if you’re on Twitter, please follow me there too. Finally, if you have math questions, feel free to send them my way via Facebook, Twitter, or by email at mathdude@quickanddirtytips.comcreate new email.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!