Determine Convergence Or Divergence Calculator

Series Convergence Tests:

\[ \text{Convergence Tests: Ratio Test, Root Test, Integral Test} \]

Convergence Result:

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1. What Is Series Convergence?

In mathematics, series convergence refers to whether the sum of an infinite sequence of terms approaches a finite limit. Determining convergence or divergence is fundamental in calculus and analysis.

2. How Do Convergence Tests Work?

Common tests include:

\[ \text{Ratio Test: } \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L \] \[ \text{Root Test: } \lim_{n \to \infty} \sqrt[n]{|a_n|} = L \] \[ \text{Integral Test: If } f(n)=a_n \text{ and } f \text{ is positive, continuous, and decreasing, then} \] \[ \sum_{n=1}^{\infty} a_n \text{ and } \int_{1}^{\infty} f(x) dx \text{ both converge or both diverge.} \]

Where:

\( a_n \) — n-th term of the series
\( L \) — Limit value
If \( L < 1 \), series converges; if \( L > 1 \), diverges; if \( L = 1 \), inconclusive

3. Importance Of Convergence Tests

Details: Convergence tests are essential for understanding the behavior of infinite series, which have applications in physics, engineering, and financial mathematics.

4. Using The Calculator

Tips: Select the test type and enter the n-th term formula of your series. The calculator will analyze convergence/divergence based on the selected test.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between absolute and conditional convergence?
A: A series converges absolutely if the sum of absolute values converges. Conditional convergence means the series converges but not absolutely.

Q2: When should I use the ratio test vs the root test?
A: Ratio test is often easier for series with factorials or exponential terms. Root test is better for series where terms are raised to the n-th power.

Q3: What are the limitations of these tests?
A: Some tests give inconclusive results (L=1). The integral test requires the function to be positive, continuous, and decreasing.

Q4: Can these tests determine the sum of a series?
A: No, convergence tests only determine whether a series converges or diverges, not the actual sum.

Q5: What about alternating series?
A: Alternating series require the Alternating Series Test, which checks if terms decrease in absolute value and approach zero.