Series Diverge or Converge Calculator

Common Convergence Tests:

\[ \text{Ratio Test: } \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L \] \[ \text{Root Test: } \lim_{n \to \infty} \sqrt[n]{|a_n|} = L \] \[ \text{Comparison Test: Compare to known series} \]

Series Type:

First Parameter:

Second Parameter (if applicable):

Test to Apply:

Result:

Explanation:

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1. What is Series Convergence?

A series converges if the sequence of its partial sums approaches a finite limit. Divergence occurs when the partial sums do not approach a finite limit. Determining convergence is fundamental in calculus and analysis.

2. How Convergence Tests Work

Common convergence tests include:

\[ \text{Ratio Test: } \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \begin{cases} < 1 & \text{converges absolutely} \\ > 1 & \text{diverges} \\ = 1 & \text{inconclusive} \end{cases} \]

Where:

Ratio Test — Best for factorials/exponentials
Root Test — Useful for nth powers
Comparison Test — Compare to known series
Integral Test — For positive decreasing functions

3. Importance of Convergence Tests

Details: Convergence tests help determine whether infinite series have finite sums, which is crucial in mathematical analysis, physics, and engineering applications.

4. Using the Calculator

Tips: Select the series type, enter required parameters, and choose a test method. The calculator will determine convergence/divergence and provide explanation.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between absolute and conditional convergence?
A: Absolute convergence means the series converges when all terms are made positive. Conditional convergence means the series converges but not absolutely.

Q2: When should I use the ratio test vs root test?
A: Ratio test is often easier when terms involve factorials or exponentials. Root test may be better when terms have nth powers.

Q3: What if a test is inconclusive?
A: Try a different test. For example, if ratio test gives L=1, try comparison or integral test.

Q4: Can a series converge to more than one sum?
A: No, if a series converges, it has a unique sum. However, rearranging conditionally convergent series can change the sum.

Q5: Are there series that converge very slowly?
A: Yes, for example the harmonic series with alternating signs converges to ln(2), but very slowly.