Even in a second-semester calculus class, determining the sum of an infinite series is beyond the scope of what you’ll typically be asked to do.
One exception to this, however, is when you’re working with an infinite geometric series, and you need to determine whether its sum is finite or infinite.
For example, if you have 1/4 + 1/8 + 1/16 … the sum of this infinite series will ultimately approach – that is, converge upon – 1/2, and that makes its sum finite.
Image source: Desmos
On the other hand, if you have 10 + 20 + 40 + 80 … the sum of the series only continues to become larger, making it infinite.
As with most other calculus equations, you can determine if the sum of a series is finite or infinite as well as the sum of a finite series by hand, and it’s not a bad idea to know how to do it this way.
For speed and ease, however, there are several different types of convergence calculators you can use which we’ll go over below.
Calculate the Sum of a Convergent Geometric series
First, there needs to be a distinction made in the way “infinite” is being used.
Both the sample series above are infinite: the first approaches 1/2 without ever reaching it – there will always be more numbers in the series but they will become smaller and smaller – while the second series is infinite as it adds more, larger numbers forever.
The key distinction here, however, is whether the sums of these infinite series are finite (converging on a certain number such as 1/2) or infinite by only becoming larger into infinity.
The basic form of a geometric series is a1 + a1*r + a1*r^2 + a1r^3 +… so that a1 is the first term and r is the common ratio.
When the value of r is between −1 and 1, you can calculate the finite sum of an infinite geometric series. If r is greater than 1, however, the sum of the series is infinite and is represented by the ∞ symbol.
Therefore, the finite sum S of a geometric series where −1 < r < 1 is determined by the formula S=a1/(1−r).
There are four steps to determine if an infinite geometric series has a finite sum and, if so, what that sum is:
- Identify the value of r from the geometric series formula.
- Determine if the series converges. That is, if the value of r is greater than one, the sum of the series is infinite.
- Find the first term by using the value of n from the geometric series formula.
- Plug in your geometric series values to the S=a1/(1−r) formula to calculate its sum.
To see an example problem solved, check out this detailed step-by-step solution.
See, that’s not complicated at all! Or to put it another way, that’s why there are a variety of convergence calculators available to do the heavy lifting for you.
As you’ll see below, there are many different online convergence calculators to choose from.
Each one, however, offers slightly different options and levels of accompanying information.
Or, if you’re using a TI-83 or TI-84 Plus, you can skip straight to the ticalc.org section to download programs to perform this function on your graphing calculator.
Image source: Symbolab
The Symbolab website provides multiple calculators to perform functions using integrals, equations, limits, tangent lines, and more.
Plus, you can use scientific notation and math symbols as opposed to just inputting numbers and text.
On the convergence calculator page, it’s simple enough to use the available math symbols to create your geometric series formula.
After that, you’ll receive not just the answer but also a step-by-step solution with accompanying explanations along the way. In addition, you can access absolute convergence and power series calculators.
With over sixty million registered users, a free account lets you store up to ten problems as well as do practice problems on a single topic.
A paid account offers additional features such as no ads, a mobile app, unlimited storage, and thousands of practice quizzes.
Image source: Wolframalpha
Wolfram|Alpha’s goal is to help people at all skill levels perform dynamic calculations on their own as opposed to hunting endlessly through the Internet for a specific answer.
In addition to its web-based tools, it also has subject-specific and reference mobile apps for the iPhone, iPad, Kindle fire, Android, and Windows.
The convergence calculator is easy enough to use and only requires numbers and text in three fields to produce both the geometric series formula and the sum for a finite series.
Unfortunately, you cannot access the steps by which the equation was performed. You can, however, customize and embed the calculator on your own web page or easily share it with others.
Image source: Emathhelp
This site also offers students many tools and resources to choose from.
Not only are there dozens of calculators organized by subject – algebra 1 and 2, calculus, linear algebra, statistics, and more – you can also access math games, logic puzzles, and answers to previously submitted math questions.
Have a question of your own? Register for free to submit it.
The series and sum calculator page gives you six options to choose from: geometric, binomial series, power, arithmetic, infinite, and partial sum.
Once again, there are four value fields you’ll enter the necessary formula info into, and after that, you’ll be provided with the geometric series formula and the answer.
In addition, there are handy instructions at the top of the page which walk you through the steps for each of the different series and sum processes available.
One note of caution: This site is dependent on advertising and if you open too many different pages at it at once, your web browser’s performance may begin to bog down.
TutorVista is primarily an online tutoring company, so the information you’ll be able to access for free is relatively limited.
If you do sign up for its services, you can receive help for both K-12 and college courses in math, statistics, chemistry, and physics and work with a live, online tutor for seven or fifteen hours a month.
As is, the convergence calculator is very barebones. You’ll plug in the necessary information as with the sites above, and you’ll receive the geometric series formula and finite sum (if available) as output.
The page does, however, provide an overview of solving this equation by hand as well as sample problems with answers you can work through to see how well you’re doing.
Image source: Desmos
Desmos provides educational support for math, primarily for grades 6-12.
It has a proprietary, web-based graphing calculator which can be used by visually impaired students as well as be embedded on external web pages. In addition, teachers can utilize the Activity Builder to create their own lessons and learning exercises.
For calculating the convergence of an infinite geometric series, a split screen is used.
The left side allows students to input the necessary equation values step-by-step while the graphical results are plotted on an x-y axis on the right-hand side of the screen.
While the options available are powerful compared to many of the sites above, it’s not very intuitive to use, and you’ll need to take a close look at the instructions and tutorials to utilize its features.
Choosing a Convergence Calculator
There’s no one right choice out of the options above.
All of them will perform the basic, necessary calculations while some provide additional bells and whistles you may not need or want to use.
You should look at each one, however, to find the one which will work best for your purposes.
Whether you need to determine the convergence – that is, the finite sum – of one infinite geometric series or will be doing it for a whole semester, these tools will make your life much easier than doing it by hand.
Need a quadratic equation calculator? Check out our guide to the best resources available to use.
*Images are screenshots by the author.