Interval of Convergence Calculator

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Interval of Convergence Lesson

Lesson Contents

What is Interval of Convergence?

For a power series, the interval of convergence is the interval in which the series has absolute convergence. It is expressed in interval notation. For example, a series that converges between 2 (inclusive) and 8 (exclusive) may be written as [2, 8) or as 2 ≤ x < 8.

A power series is an infinite series of the form:

$$\sum\limits_{n = 0}^\infty {{c_n}{{\left( {x – a} \right)}^n}}$$

Where cn is a coefficient that varies with n and the series is a function of x with its terms varying with the nth term of the series.

Let’s look further into the meaning of convergence in the context of a power series. A power series adds an infinite number of successive terms. The sum of those terms can either be finite or infinite.

The series converges if the sum of those terms is a finite number. The series diverges if the sum of those terms is infinite. By solving for the interval of convergence, we find the range of values for x in |x – a| < R such that the series converges.

Calculating the Interval of Convergence of a Power Series

When solving for convergence of a power series, we have multiple test options to choose from. These include the very common ratio test and root test. Since the ratio test is user-friendly and used by the calculator on this page, we learn how to use it here.

With the ratio test, we use a ratio of the power series and a modified n + 1 version of itself to solve for the values of x that satisfy the convergence criteria. The formula for the ratio test is:

$$\text{Convergence when} \; L < 1, \; L = \lim_{n\to\infty} \left\lvert\frac{a_{n+1}}{a_{n}} \right\rvert$$

Where an is the power series and an + 1 is the power series with all terms n replaced with n + 1.

The first step of the ratio test is to plug the original and modified versions of the power series into their respective locations in the formula. We may simplify the resulting fraction.

Then, evaluate the limit as n approaches infinity. By plugging infinity in for n, the expression may become what appears to be unsolvable. We then start cancelling out terms that are insignificant compared to infinity and eliminate the actual infinity terms from the expression.

After evaluating the limit and simplifying the resultant expression, set up the expression such that L < 1. We now have an inequality resembling the form of 1c×|x – a| < 1. The constant c can be fractional or non-fractional.

Solve for the left and right endpoint that satisfy the final inequality. Those are the interval of convergence bounds. We must determine if each bound is inclusive or exclusive. To do this, we check for series convergence/divergence at those points.

Plug the left endpoint value x = a1 in for x in the original power series. Then, take the limit as n approaches infinity. If the result is nonzero or undefined, the series diverges at that point. Divergence indicates an exclusive endpoint and convergence indicates an inclusive endpoint. Repeat the process for the right endpoint x = a2 to complete the interval of convergence.

How the Calculator Works

This interval of convergence calculator is primarily written in JavaScript (JS). Because the computation routine is JS, it runs entirely in your browser in real-time. This allows near-instant solutions and avoids the usual page reloads seen on other calculator websites.

The computation routine also utilizes a JS-native computer algebra system (CAS). The CAS performs various symbolic operations throughout the routine, such as polynomial division and limit evaluation.

The routine itself is exactly the same as explained in this lesson. It uses the ratio test by filling out the formula with your inputted power series. Various states of the expression are saved along the way and used for the solution steps. The answer and solution steps are procedurally built out and rendered as LaTeX code (a math rendering language).

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