Series Convergence Calculator

Category: Calculus

- June 12, 2025

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Determine whether a mathematical series converges or diverges and calculate its sum (when applicable) using various convergence tests.

Series Input

Series Type:

General Term a_n:

Use 'n' as the index variable. Examples: 1/n^2, (2^n)/n!, 1/(n*log(n))

Starting Index:

The first value of n in the summation

Number of Terms to Approximate:

For numerical approximation of the sum

Test Selection

Ratio Test

Root Test

Comparison Test

Integral Test

Alternating Series Test

Understanding Series Convergence

An infinite series is the sum of infinitely many terms. A series converges if the sequence of its partial sums approaches a finite limit; otherwise, it diverges.

Common Series Types

p-Series: Σ 1/n^p converges when p > 1, diverges when p ≤ 1
Geometric Series: Σ ar^(n-1) converges to a/(1-r) when |r| < 1, diverges when |r| ≥ 1
Harmonic Series: Σ 1/n is a special case of p-series with p = 1, and it diverges
Alternating Series: Σ (-1)^(n+1)·a_n may converge even when Σ a_n diverges
Telescoping Series: Series where terms cancel each other, often leaving only a few terms

Key Convergence Tests

Ratio Test: If lim|a_(n+1)/a_n| = L < 1, the series converges absolutely
Root Test: If lim|a_n|^(1/n) = L < 1, the series converges absolutely
Comparison Test: Compare your series with a known convergent/divergent series
Integral Test: For positive decreasing terms, compare series with improper integral
Alternating Series Test: If terms decrease in absolute value and approach zero, series converges

Important Series Results

Riemann Zeta Function: ζ(2) = Σ 1/n² = π²/6 ≈ 1.6449
Basel Problem: Special case of Riemann zeta function with p = 2
Alternating Harmonic Series: Σ (-1)^(n+1)/n = ln(2) ≈ 0.6931
Geometric Series Sum: When |r| < 1, Σ ar^(n-1) = a/(1-r)
Convergence Speed: Higher values of p in p-series lead to faster convergence

Disclaimer: This calculator provides analytical results for common series types and uses numerical approximations for partial sums. For complex series, the tests may not be conclusive. Always verify important results with formal mathematical proofs.

General Form of a Series:

$$ \sum_{n=1}^{\infty} a_n $$

Examples:

p-Series: $$ \sum_{n=1}^{\infty} \frac{1}{n^p} $$
Geometric Series: $$ \sum_{n=1}^{\infty} ar^{n-1} $$
Alternating Series: $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^p} $$

What Is the Series Convergence Calculator?

The Series Convergence Calculator is an interactive tool that helps you determine whether an infinite mathematical series converges to a finite value or diverges. It supports a variety of series types, such as p-series, geometric series, harmonic series, alternating series, and telescoping series. If the series converges, the calculator provides an estimate of its sum using numerical approximation and analytical insights.

Why Use This Calculator?

Understanding series convergence is important in Calculus, mathematical analysis, and applications across Physics, engineering, and economics. This calculator simplifies that process by offering:

Instant results for common series types
Step-by-step convergence tests like the Ratio Test and Root Test
Graphical visualisation of terms and partial sums
LaTeX-style mathematical formulas for clarity

It complements tools like a Partial Derivative Calculator, Antiderivative Calculator, and Limit Calculator for students and professionals working with series, differentiation, and integration.

How to Use the Calculator

Select the Series Type from the dropdown menu (e.g., p-Series, Geometric, Custom).
Enter the required parameters such as the value of p, general term, or ratio depending on the type.
Set the Starting Index and the Number of Terms for approximation.
Choose one or more Convergence Tests to apply.
Click the Analyze Series button to get the result.

Features and Outputs

Summary Result: Tells you if the series converges or diverges.
Approximate Sum: Provided when the series converges.
Convergence Tests: Includes Ratio Test, Root Test, Integral Test, and more.
Graph: Visualises the behaviour of individual terms and partial sums.
Formula Display: Shows the symbolic form of the series.

Helpful for Learning and Exploration

Whether you’re studying for exams or exploring mathematical series, this tool enhances your understanding through visualisation and structured analysis. It pairs well with tools such as the Integral Calculator for definite or indefinite integration, the Second Derivative Calculator for analysing curve behaviour, and the Interval of Convergence Calculator for power series evaluations.

Frequently Asked Questions

What does it mean for a series to converge?
A series converges if the sum of its terms approaches a fixed number as more terms are added. Otherwise, it diverges.

Can this tool handle custom series?
Yes. Enter a valid general term using n as the index. Examples: 1/n^2, (2^n)/n!.

How accurate are the results?
The calculator uses up to 10,000 terms for numerical approximation. Results are reliable for most common series, but for complex expressions, mathematical proof is recommended.

What if I want to analyse multivariable functions?
Use related tools like the Partial Derivative Calculator or Tangent Plane Calculator to compute partial derivatives and surface approximations.

Conclusion

The Series Convergence Calculator is a practical resource for checking convergence, understanding series behaviour, and estimating sums. It makes mathematical analysis more intuitive and supports deeper insights into functions, just like tools for finding derivatives, solving integrals, or evaluating limits.