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Convergence Test Calculator + Online Solver With Free Steps
The Convergence Test Calculator is used to find out the convergence of a series. It works by applying a bunch of Tests on the series and finding out the result based on its reaction to those tests. Calculating the sum of a Diverging Series can be a very difficult task, and so is the case for any series to identify its type. So, certain tests have to apply to the Function of the series to get the most appropriate answer.What Is a Convergence Test Calculator?
The Convergence Test Calculator is an online tool designed to find out whether a series is converging or diverging. The Convergence Test is very special in this regard, as there is no singular test that can calculate the convergence of a series. So, our calculator uses several different testing methods to get you the best result. We will take a deeper look at them as we move forward in this article.How to Use the Convergence Test Calculator?
To use the Convergence Test Calculator, enter the function of the series and the limit in their appropriate input boxes and press the button, and you have your Result. Now, to get the stepbystep guide to making sure you get the best results from your Calculator, look at the given steps:Step 1
We start by setting up the function in the appropriate format, as the variable is recommended to be n instead of any other. And then enter the function in the input box.Step 2
There are two more input boxes, and these are the ones for “to” and “from” limits. In these boxes, you are to enter the lower limit and the upper limit of your series.Step 3
Once, all of the above steps are completed, you may press the button labeled “Submit”. This will open up a new window where your solution will be provided.Step 4
Finally, if you wish to find out about more series’ convergence you can enter your new problems in the new window, and get your results.How Does the Convergence Test Calculator Work?
The Convergence Test Calculator works by testing a series to the limit of infinity and then concluding whether it’s a Convergent or Divergent series. This is important because a Convergent Series will converge to a certain value at some point at infinity, and the more we add the values into such a series the closer we get to that Certain Value. Whilst, on the other hand, Divergent Series do not get a defined value as you add them, they instead diverge either into infinity or some random sets of values. Now, before we move forward to discuss how to find the Convergence of a series, let’s first discuss what a series is.Series
A Series in mathematics is referred to as a process rather than a quantity, and this Process involves adding a certain function to its values again and again. So, a series at its core is indeed a polynomial of some kind, with an Input variable that leads to an Output value. If we apply a Summation function on top of this polynomial expression, we have a series limits of which are often approaching Infinity. So, a series could be expressed in the form: \[ \sum_{n=1}^{\infty} f(n) = x \] Here, the f(n) describes the function with variable n and the output x could be anything from a defined value to Infinity.Convergent and Divergent Series
Now, we will investigate what makes a series Convergent or Divergent. A Convergent Series is one that when added up many times will result in a particular value. This value can be approached as a value of its own, so let our Convergent Series result in a number x after 10 iterations of the summation. Then, after 10 more it will approach a value that would be not too far from x but a better approximation of the series’ result. An Important Fact to notice is that the result from more sums would be almost always Smaller than the one from lesser sums. A Divergent Series on the other hand when added more times would usually result into a bigger value, which would keep increasing thus diverging that it would approach Infinity. Here, we have an example of each Convergent as well as Divergent Series: \[ Convergent: \phantom {()} \sum_{n=1}^{\infty} \frac {1} {2^n} \approx 1 \] \[ Divergent: \phantom {()} \sum_{n=1}^{\infty} 112 n \approx \infty \]Convergence Tests
Now, to test the convergence of a series, we can use several techniques called Convergence Tests. But it must be noted that these tests only come into play when the Series’ Sum cannot be calculated. That occurs very commonly when dealing with values adding up to Infinity. The first test we look at is called the Ratio Test.
Ratio Test

Root Test