Series Convergent Or Divergent Calculator

Series Convergence Tests:

Common tests include:

Ratio Test: \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \)
Root Test: \( \lim_{n \to \infty} \sqrt[n]{|a_n|} \)
Comparison Test
Integral Test

Result:

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1. What is Series Convergence?

A series is the sum of the terms of a sequence. A series is convergent if the sequence of its partial sums approaches a finite limit. Otherwise, it is divergent.

2. How Does the Calculator Work?

The calculator applies standard convergence tests based on the series type:

Geometric Series: \( \sum_{n=0}^{\infty} ar^n \) converges if \( |r| < 1 \)
p-Series: \( \sum_{n=1}^{\infty} \frac{1}{n^p} \) converges if \( p > 1 \)
Alternating Series: \( \sum_{n=1}^{\infty} (-1)^n b_n \) converges if \( b_n \) decreases and \( \lim b_n = 0 \)

3. Importance of Convergence Tests

Details: Determining series convergence is fundamental in calculus and analysis, with applications in physics, engineering, and probability.

4. Using the Calculator

Tips: Select the series type and enter the appropriate parameter (common ratio for geometric series, exponent for p-series, etc.).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between absolute and conditional convergence?
A: A series is absolutely convergent if the series of absolute values converges. Conditionally convergent series converge but not absolutely.

Q2: Can a series converge to zero?
A: The terms must approach zero for convergence, but the sum itself approaches a finite limit, not necessarily zero.

Q3: What about more complex series?
A: This calculator handles basic cases. Complex series may require ratio test, root test, or comparison tests.

Q4: Why does the harmonic series diverge?
A: The harmonic series \( \sum \frac{1}{n} \) is a p-series with p=1, which is the boundary case for divergence.

Q5: Can series convergence help with integrals?
A: Yes, the Integral Test relates the convergence of series to improper integrals.