The radius of Convergence Calculator is a free tool that displays the convergence point for any given series. STUDYQUERIES’s online radius of convergence calculator makes calculations faster, and it displays the convergence point in a fraction of a second.

## How to Use the Radius of Convergence Calculator?

• Step 1: Enter the function and range in the appropriate input fields
• Step 2: Now click “Calculate” in order to calculate the results
• Step 3: In the new window, you will find the convergence point for the given series

## What is the Radius of Convergence?

When you are a first-year college student, you are taught infinite series and the radius of convergence. The infinite series is a polynomial in a variable X with an infinite number of terms.

$$f(X) = K_0x^0 + K_1x^1 + K_2x^2 + K_3x^3 + …$$

In particular, if all of the Ki are treated as one, we can examine the series,
$$f(X) = 1 + x^1 + x^2 + x^3 + …$$
If we pick X=1/2, we have to add up smaller and smaller terms

$$1 + 1/2 + 1/4 + 1/8 + 1/16 + …$$

Each number is half the size of the previous. Every number takes you half the remaining distance to two: the sum is said to converge to two.

It isn’t always possible to get a reasonable answer by adding up an infinite number of numbers. We can get the sum of all the Ki, where X=2, assuming all the Ki are one:

$$1 + 2 + 4 + 8 + 16 …$$

Clearly, this becomes infinite. Normally, we say diverges because the sum diverges, rather than converges to infinity). If we set X=-2, we get a worse problem:

$$1 – 2 + 4 – 8 + 16 …$$

In order to add the numbers on the left, we get 1, then -1, then 3, then -5, then 11, we flip-flop between larger and larger positive and negative numbers.

With a little effort, you can convince yourself that the infinite sum doesn’t make sense if X is bigger than one or smaller than minus one (it diverges). The sum can be done for X smaller than one and bigger than minus one. This function has a radius of convergence of one.

We should mention four more things.

Mathematicians call it a radius of convergence because they like to plug in complex numbers, like X = U + i V, where i * i = -1. They can show that the series converges as the radius of convergence increases U² + V² = R², and diverges outside the circle.

In fact, you can deduce that our sum (with all Ki=1) will always give 1/(1-X) when it converges. The series represents a function in this sense. Taylor’s theorem explains how to get the series from the function. Other infinite series Ki can also be used to make other functions.

This sum $$1/(1-X)$$ does become big when X gets close to X=1. In contrast, there doesn’t seem to be anything wrong at X=-1. Due to a problem at X=1, which happens to be at the same distance from zero, the convergence of the infinite series at X=-1 is spoiled! It is a distance from the point where the function blows up or becomes strange where the radius of convergence is determined.

Calculating the radius of convergence for a series Ki is a simple process (the ratio test). It is impossible for the series to converge unless the terms eventually get smaller and smaller. If we insist that

$$|K_n+1 X_n+1|$$

be smaller than

$$|K_n X_n|,$$

that can happen only if

$$|X|$$

is smaller than

$$\|K_n/K_n+1|.$$

The radius of convergence of an infinite series is (basically) the value of

$$|K_n/K_n+1|$$

for large n.

### Interval of Convergence

The interval of convergence of a power series: $$\int_{n=0}^{\infty}C_n(x-a)^n$$

X-value interval within which convergent series can be plugged into the power series. A power series’ anchor point, a, is always the center of the interval of convergence.

• There is a positive number R such that the series diverges for x with $$|x−a| > R$$ but converges absolutely for x with $$|x − a| < R.$$ The series may or may not converge at either of the endpoints x = a − R and x = a + R.
• (R = ∞) The series converges absolutely for every x
• At x = a, the series converges and diverges elsewhere (R = 0)

A power series’ interval of convergence is the largest interval I such that for any value of x in I the power series converges. Knowing the radius of convergence can be used to calculate the interval of convergence. First, you solve the inequality $$|x − a| < R$$ for x and then you check each endpoint individually. But how do we calculate the radius of convergence? We take the ratio test (or root test) and solve the problem.

### Example Of Radius of Convergence

Geometric Power Series: If all coefficients are 1, then the power series centred at x = 0 gives the geometric power series:

$$\int_{n=0}^{\infty}X^n= 1 + x + x^2 + x^3 + .. + x^n$$

This is the geometric series with first term 1 and ratio x

$$S_n = 1 + x + x^2 + x^3 + x^4 + .. + x^n$$

$$\left(x-1 \right)S_n = \left(x-1 \right)(1 + x + x^2 + x^3 + x^4 + .. + x^n)$$

$$= (1 + x + x^2 + x^3 + x^4 + .. + x^n) – (x + x^2 + x^3 + x^4 + .. + x^{n+1})$$

$$= (1-x^{n+1})$$

$$S_n = \frac{(1-x^{n})}{(1-x)}$$

So, $$\int_{n=0}^{\infty}X^n = \lim_{x \to \infty}S_n =\lim_{x \to \infty}\frac{(1-x^{n})}{(1-x)}$$

which converges if and only if |x| < 1

## FAQs

How do you find the radius of convergence?

The radius of convergence is half the length of the convergence interval. If the radius of convergence is R, then the interval of convergence includes the open interval: (a − R, a + R). Ratio Tests are used to find the radius of convergence, R.

What is the radius of convergence of the series X^n?

Regardless of the choice of x, the series converges with the Ratio Test. Therefore the radius of convergence is R=∞, and the interval of convergence is (−∞,∞).

Why is it called a radius of convergence?

It’s called a radius of convergence because mathematicians like to plug in complex numbers, like X = U + i V, where i * i = -1. Using this series, they can demonstrate that $$U^2 + V^2 = R^2$$ converges inside the circle, but diverge outside it.

Can the radius of convergence be 0?

The distance between the center of a power series’ convergence interval and its endpoints. The radius of convergence is zero if the series only converges at a single point.

What is the radius of convergence of a power series?

In mathematics, the radius of convergence of a power series is the radius of the largest disk in which the series converges.

How do you test for convergence?

• If the limit of a[n]/b[n] is positive, then the sum of a[n] converges if and only if the sum of b[n] converges.
• If the limit of a[n]/b[n] is zero, and the sum of b[n] converges, then the sum of a[n] also converges.
• If the limit of a[n]/b[n] is infinite, and the sum of b[n] diverges, then the sum of a[n] also diverges.

What is meant by the term convergence?

A process of convergence, especially the movement towards a point of union; the convergence of three rivers, particularly: coordinated movement between the two eyes allowing a single point image to be formed on their retinas. Convergence is the state or property of being convergent.

What is the radius of convergence of a geometric series?

If a power series converges only for x = a, then the radius of convergence is defined to be R = 0. – If the power series converges for all values of x, then the radius of convergence is defined to be R = ∞.

What is the radius of convergence of the Maclaurin series for?

The radius of convergence for a Maclaurin series can be found by checking which of the three situations we’re in. If our Maclaurin series converges for all real values of 𝑥, we say that our radius of convergence 𝑅 is equal to ∞.