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Interval of Convergence Calculator + Online Solver With Free Steps

The online Interval of Convergence Calculator helps you find the convergence points of a given series.

The Interval of Convergence Calculator is an influential tool mathematicians use to find the convergence points in a power series quickly. The Interval Convergence Calculator also helps you solve other complex mathematical problems.

What Is an Interval of Convergence Calculator?

An Interval Convergence Calculator is an online tool that instantaneously finds the converging values in a power series.

The Interval Convergence Calculator requires four inputs. The first input is the function you need to calculate. The second input is the name of the variable in the equation. The third and fourth inputs are the range of numbers that are required.

The Interval Convergence Calculator displays the converging points in a fraction of a second.

How To Use an Interval of Convergence Calculator?

You can use the Interval of Convergence Calculator by plugging the mathematical function, variable, and range into their respective boxes and simply clicking the “Submit” button. You will be presented with the results immediately.

The step-by-step instructions on how to use an Interval of Convergence Calculator are given below:

Step 1

First, we plug the function we are provided with into the “Enter the function” box.

Step 2

After entering the function, we input the variable.

Step 3

After entering the variable, we input the starting value of our function.

Step 4

Finally, we enter the ending value of our function.

Step 5

After plugging in all the inputs, we click the “Submit” button which calculates the points of convergence and displays them in a new window.

How Does an Interval Convergence Calculator Work?

The Interval of Convergence Calculator works by calculating the convergence points of a power series using the function and limits. The interval of convergence calculator then provides a relationship between the equation and the variable x representing the convergence values.

What Is Convergence?

In mathematics, convergence is the feature of a particular infinite series and functions of getting closer to a limit when a function’s input (variable) changes in value or as the number of terms in the series grows.

For example, the function $ y = \frac{1}{x} $ converges to zero when x is increased. However, no value of x allows the function y to become equal to zero. When the value of x approaches infinity, the function is said to have converged.

What Is a Power Series?

Power series is a series that is also known as an infinite series in mathematics and can be compared to a polynomial with an endless number of terms, such as $1 + x + x^{2} + x^{3} +…,$. 

A given power series will often converge (when it reaches infinity) for all values of x in a range near zero– particularly, If the radius of convergence, which is denoted by the positive integer r (known as the radius of convergence), is less than the absolute value of x.

A power series can be written in the following form:

\[ \sum_{n=0}^{\infty} = c_{n}(x-a)^{n} \]

Where $a$ and $c_{n}$ are numbers. The $c_{n}$ is also referred to as the coefficients of the power series. A power series is first identifiable because it is a function of x.

A power series may converge for some values of x and diverge for other values of x because the terms in the series involve the variable x. The value of the series at x=a for a power series centered at x=a is given by $c_{0}$. A power series, therefore, always converges at its center.

However, most power series converge for various values of x. The power series then either converges for all real numbers x or converges for all x within a defined interval.

Properties of Convergence In a Power Series

Convergence in a power series has several essential properties. These properties have helped mathematicians and physicists make several breakthroughs throughout the years.

A power series diverges outside the symmetric interval in which it converges absolutely around its expansion point. The distance from the endpoint and expansion point is called the radius of convergence.

Any combination of convergence or divergence may occur at the endpoints of the interval. In other words, the series may diverge at one endpoint and converge at the other, or it may converge at both endpoints and diverge at one.

The power series converges to its expansion points. This set of points where the series connect is known as the interval of convergence.

Why Are Power Series Important?

Power series are important because they are essentially polynomials; they are more convenient to use than most other functions such as trigonometric and logarithms, and they help compute limits and integrals as well as solve differential equations.

Power series have the characteristic that the more terms you add up, the closer you are to the precise sum. Computers frequently use them to approximate the value of transcendental functions because of this feature. By adding some elements in an infinite series, your calculator provides a close approximation of sin(x).

Sometimes it is helpful to allow the first few terms of the power series to act as a stand-in for the function itself rather than utilizing the power series to approximate a specific value of a function.

For instance, in a differential equation, they could not typically solve, students in first-year physics studies are instructed to substitute sin(x) with the first term of its power series, x. Power series are used in a similar way throughout physics and mathematics.

What Is an Interval of Convergence?

Interval of Convergence is the series of values for which a sequence converges. Just because we can identify an interval of convergence for a series doesn’t entail that the series as a whole is convergent; instead, it just means that the series is convergent during that particular interval.

For example, imagine that the interval convergence of a series is -2 < x < 8. We graph a circle around the endpoints of the series along the $ x \ axis $. This allows us to visualize the interval of convergence. The diameter of the circle can represent the interval of convergence.

The following equation is used to find the interval of convergence:

\[ \sum_{n=0}^{\infty} = c_{n}(x-a)^{n} \]

The interval of convergence is represented in the following manner:

 a < x < c 

What Is a Radius of Convergence?

The radius of convergence of a power series is the radius that is half the value of the interval of convergence. The value can either be a non-negative number or infinity. When it is positive, the power series thoroughly and evenly converges on compact sets within the open disc with a radius equal to the radius of convergence.

If a function has several singularities, the radius of convergence is the shortest or most diminutive of all the estimated distances between each singularity and the center of the convergence disc. 

$R$ represents the radius of convergence. We can also form the following equation:

 (a-R, a + R) 

How To Calculate the Radius and Interval of Convergence

To calculate the radius and interval of convergence, you need to perform a ratio test. A ratio test determines whether a power series can converge or diverge. 

The ratio test is done using the following equation:

\[ L = \lim_{n \to \infty} \left | \frac{a_{n+1}}{a_{n}} \right | \]

If the ratio test is L < 1, the series is converging. A value of L > 1 or L = $\infty $ means that the series is diverging. The test becomes inconclusive if L = 1 . 

Assuming we have a series with L < 1 we can find the radius of convergence (R) by the following formula:

\[ \left | x – a  \right | < R \] 

We can also find the interval of convergence by the equation written below:

a – R < x < a + R 

After obtaining the interval of convergence, we must verify the convergence of the interval’s endpoints by inserting them into the initial series and using any available convergence test to determine whether or not the series converges at the endpoint.

If a power series diverges from both ends, the interval of convergence would be as follows:

a – R < x < a + R 

If a series diverges on its left side, the interval of convergence can be written as:

\[ a – R < x \leq  a + R \]

And finally, if the series diverges to the right endpoint, the interval of convergence would be as follows:

\[ a – R \leq x < a + R \]

This is how radius and interval of convergence are calculated.

Solved Examples

The Interval of Convergence Calculator can easily find the converging points in a power series. Here are some examples that were solved using the Interval of Convergence Calculator.

Example 1

A high school student is given a power series equation $ \sum_{n=1}^{\infty}\frac {n(x-4)^n}{3^n} $. The student needs to check if the power series converges or not. Find the Interval of Convergence of the given equation.

Solution

We can easily find the interval of convergence by using the Interval of Convergence Calculator. First, we plug in the equation in the equation box. After entering the equation, we plug in our variable letter. Finally, in our case, we add our limit values 0 and $ \infty $.

Finally, after entering all our values, we click the “Submit” button on the Interval of Convergence Calculator. The results are displayed immediately in a new window.

Here are the following results we get from the Interval of Convergence Calculator:

\[ \sum_{n=1}^{\infty}\frac {n(x-4)^n}{3^n} \ \ converges \ when \left | x-4 \right |<3 \]

Example 2

During his research, a mathematician needs to find the interval of convergence of the following equation:

\[ \sum_{n=1}^{\infty}\frac {n(x+5)^n}{4^n} \]

Using the Interval of Convergence Calculator, find the Interval of convergence.

Solution

Using the Interval of Convergence Calculator, we can easily calculate the points where the series converge. First, we input the function into its respective box. After inputting the process, we declare a variable we are going to use; we use $n$ in this case. After expressing our variable, we input the limit values, which are 0 and $\infty$.

Once we have inputted all our initial variables and functions, we click the “Submit” button. The results are created instantaneously in a new window. The Interval of Convergence Calculator gives us the following results:

\[ \sum_{n=1}^{\infty}\frac {n(x+5)^n}{4^n} \ \ converges \ when \left | x+5 \right |<4 \]

Example 3

While solving an assignment, a college student comes across the following power series function:

\[ \sum_{n=1}^{\infty}\frac {n(4x+8)^n}{2^n} \]

The student must determine if this power series converges to a single point. Find the interval of convergence of the function.

Solution

The function can easily be solved using the Interval of Convergence Calculator. First, we enter the function provided to us in the input box. After the function is entered, we define a variable, $n$, in this case. Once we plug in the function and variable, we enter the limits of our function, which are $1$ and $\infty$. 

After entering all the values in the Interval of Convergence Calculator we click the “Submit” button and the results are displayed in a new window. The Interval of Convergence Calculator gives us the following result:

\[ \sum_{n=1}^{\infty}\frac {n(4x+8)^n}{2^n} \ \ converges \ when \left | 4x+8 \right |<2 \]

Example 4

Consider the following equation:

\[ \sum_{n=1}^{\infty}\frac {n(10x+20)^n}{5^n} \]

Using the equation above, find the interval of convergence in the series.

Solution

We will solve this function and calculate the interval of convergence using the Interval of Convergence Calculator. We will simply enter the function in its respective box. After entering the equation, we assign a variable $n$. After performing these actions we set the limits for our function, which are n=1 to $n = \infty$.

Once we have plugged in all out initial values we click the “Submit” button, and a new window with the answer will be displayed. The result from the Interval of Convergence Calculator is shown below:

\[ \sum_{n=1}^{\infty}\frac {n(10x+20)^n}{5^n} \ \ converges \ when \left | 10x+20 \right |<5 \]

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