Series Calculator
This calculator tests the convergence or divergence of infinite series and computes partial sums. Enter your series to analyze convergence, find the sum, and view step-by-step solutions.
Series Analysis Results
Enter a series expression to analyze its convergence and compute partial sums.
About Series Convergence
An infinite series is the sum of the terms of an infinite sequence. Determining whether a series converges (approaches a finite value) or diverges (does not approach a finite value) is a fundamental concept in calculus and mathematical analysis.
Common Series Tests
The calculator applies several tests to determine convergence:
- Divergence Test: If the limit of terms doesn't approach zero, the series diverges.
- Integral Test: Compares the series to an improper integral.
- Comparison Test: Compares to a known convergent or divergent series.
- Ratio Test: Useful for series with factorials or exponentials.
- Root Test: Similar to ratio test but uses nth roots.
- Alternating Series Test: For series with alternating signs.
Types of Series
Common series types include:
- Geometric Series: ∑arⁿ converges if |r| < 1, sum is a/(1-r)
- p-Series: ∑1/nᵖ converges if p > 1
- Telescoping Series: Many terms cancel out when expanded
- Harmonic Series: ∑1/n diverges despite terms approaching zero
- Alternating Harmonic Series: ∑(-1)ⁿ⁺¹/n converges conditionally
Applications
Series convergence is essential in:
- Taylor and Maclaurin series expansions
- Fourier series for periodic functions
- Power series solutions to differential equations
- Numerical approximations in physics and engineering
- Probability theory and statistics
Limitations
While this calculator handles many common series, some complex series may require manual analysis or specialized tests not implemented here.