Interval of Convergence Calculator with Steps | GoMim Math AI

What is Interval of Convergence?

The interval of convergence is the set of all values for which a power series converges.

A power series has the form:

$$\sum_{n=0}^{\infty} a_n (x-c)^n$$

where $a_n$ are constants, $x$ is the variable, and $c$ is the center. The series converges for certain values of $x$ and diverges for others. Determining the interval of convergence often involves calculating the radius of convergence $R$ and testing the endpoints. Using an Interval of Convergence Calculator can simplify this process and ensure accurate results.

How to Calculate Interval of Convergence Step-by-Step

Finding the interval of convergence can be done manually or using a calculator. Follow these steps using the ratio test:

1、Identify the Series: Consider a power series $\sum_{n=0}^{\infty} a_n (x-c)^n$.

2、Apply the Ratio Test: Compute

  $$L = \lim_{n \to \infty} \left| \frac{a_{n+1}(x-c)^{n+1}}{a_n(x-c)^n} \right| = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \cdot |x-c|$$

3、Determine Convergence: Solve the inequality $L < 1$ to find values of $x$.

4、Find Radius of Convergence $R$: Solve $|x-c| < R$.

5、Check Endpoints: Test $x = c+R$ and $x = c-R$ for convergence using appropriate tests (e.g., alternating series or p-series test).

Example 1:

Series: $\sum_{n=0}^{\infty} \frac{(x-3)^n}{n+1}$

Applying the ratio test:

$$\lim_{n \to \infty} \left| \frac{(x-3)^{n+1}}{n+2} \cdot \frac{n+1}{(x-3)^n} \right| = |x-3| < 1$$

Interval: $2 < x < 4$

Example 2:

Series: $\sum_{n=0}^{\infty} \frac{n(x-2)^n}{3^n}$

Ratio test:

$$L = \lim_{n \to \infty} \left| \frac{(n+1)(x-2)^{n+1}/3^{n+1}}{n(x-2)^n/3^n} \right| = |x-2| < 3$$

Radius: $R = 3$, so interval is $-1 < x < 5$

Check endpoints with a power series interval of convergence calculator if needed.

Related Practice Problem

Problem 1: Determine the interval of convergence for

$$\sum_{n=1}^{\infty} \frac{(x+1)^n}{n^2}$$

Step-by-Step Solution:

1、Identify the series.

2、Apply ratio test:

  $$L = \lim_{n \to \infty} \left| \frac{(x+1)^{n+1}}{(n+1)^2} \cdot \frac{n^2}{(x+1)^n} \right| = |x+1|$$

3、Solve for convergence: $|x+1| < 1 \Rightarrow -2 < x < 0$

4、Check endpoints:

• $x=-2$: $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$ converges (alternating series test)
• $x=0$: $\sum_{n=1}^{\infty} \frac{1}{n^2$}}\converges (p-series test)

Conclusion: Interval of convergence: $[-2, 0]$

Problem 2: Find the interval of convergence for

$$\sum_{n=1}^{\infty} \frac{(2x-1)^n}{n\cdot 5^n}$$

Step-by-Step Solution:

1、Ratio test:

$$L = \lim_{n \to \infty} \left| \frac{(2x-1)^{n+1}/[(n+1)5^{n+1}]}{(2x-1)^n/(n5^n)} \right| = \frac{|2x-1|}{5} < 1$$

2、Solve: $|2x-1| < 5 \Rightarrow -2 < x < 3$

3、Check endpoints with a radius of convergence calculator for accurate inclusion.

Conclusion: Interval of convergence: $(-2, 3)$

Why Interval of Convergence is Important

The interval of convergence is essential because it determines the range of xxx values where a power series converges and can be applied reliably. In calculus and mathematical analysis, knowing the interval ensures that power series approximations of functions remain accurate. Engineers and scientists use this concept to model real-world phenomena, solve differential equations, and analyze systems such as signal processing or mechanical vibrations. In data analysis, power series can assist in regression modeling and forecasting trends. Using an Interval of Convergence Calculator helps students, researchers, and professionals quickly verify convergence ranges, ensuring both academic accuracy and practical reliability.

Use GoMim Math AI Solver for Interval of Convergence

The GoMim Math AI Solver acts as a smart Interval of Convergence Calculator, simplifying the computation of convergence intervals for any power series. By inputting your series into GoMim, you can instantly determine the radius of convergence, test endpoints, and find the complete interval of convergence with step-by-step explanations. This saves time, reduces manual errors, and makes the process accessible to students, educators, and professionals. Whether for homework, research, or engineering applications, GoMim ensures accurate and efficient results. Try it now and experience how quickly power series problems can be solved.

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FAQ

Q: What is the interval of convergence?

A: It is the range of values for which a power series converges to a finite sum.

Q: How do you find the interval of convergence?

A: You typically use tests such as the ratio test or root test to find the interval, followed by checking the endpoints for convergence.

Q: Why is the interval of convergence important?

A: It ensures that power series can be used effectively in mathematical modeling and calculations without diverging.

Q: Can the interval of convergence be infinite?

A: Yes, if the series converges for all real numbers, the interval is infinite, denoted as \((- \infty, \infty)\).

Q: What happens if a series diverges outside its interval of convergence?

A: Outside the interval, the series does not sum to a finite value, making it unreliable for calculations.

Q: Can AI tools calculate intervals of convergence?

A: Yes, AI tools like GoMim Math AI Solver can efficiently compute convergence intervals for complex series.