# Series Convergence Tester (Ratio/Root/Integral/Comparison)

Numeric convergence tests for Σ aₙ: nth-term, ratio, root, p-series comparison, and integral test; alternating series classified by Leibniz with a sum estimate.

> Canonical page: https://elysiatools.com/en/tools/series-convergence-tester

- **Category:** Math & Numbers

- **Keywords:** series convergence, ratio test, root test, integral test, comparison test, p-series, harmonic series, alternating series, leibniz test, calculus

## Overview

The Series Convergence Tester evaluates the convergence behavior of infinite series Σ aₙ using standard calculus tests. It analyzes user-defined terms with the nth-term divergence test, ratio test, root test, p-series comparison, integral test, and Leibniz alternating series test, providing numeric results and sum approximations.

## Inputs

- **Term aₙ expression (variable n)** (text): The general term aₙ as an expression of n (n starts at 1). Supports + - * / ^, parentheses, and functions like exp, sin, cos, sqrt, ln. Alternating signs (e.g. cos(pi*n)) are detected.
- **Decimal Places** (number)

## When to use

- Verifying homework or exam solutions involving infinite series convergence in calculus courses.
- Classifying series as absolutely convergent, conditionally convergent, or divergent across multiple mathematical criteria.
- Estimating the infinite sum of convergent series alongside tail approximations.

## How it works

- Enter the general term aₙ as an algebraic expression of n (e.g., 1/n^2, cos(pi*n)/n, or 0.5^n).
- Set the desired precision by adjusting the Decimal Places option (from 0 to 8).
- Review the automated suite of convergence tests, including limits for the ratio (L), root (ρ), p-series exponent, integral estimates, alternating sign detection, and the calculated sum estimate.

## Use cases

- Testing whether a power-law term like 1/n^p converges by evaluating the estimated exponent p against 1.
- Differentiating between absolute and conditional convergence on alternating harmonic series.
- Computing rapid numeric approximations for infinite series sums with customizable decimal precision.

## Frequently asked questions

### What mathematical functions can I use in the expression for aₙ?

You can use standard arithmetic operators (+, -, *, /, ^), parentheses, and functions like exp, sin, cos, sqrt, and ln with variable n.

### How does the tool handle alternating series?

It detects alternating sign patterns such as cos(pi*n) or (-1)^n, checks if magnitudes decrease to zero, and applies the Leibniz alternating series test to identify conditional convergence.

### Why do the ratio and root tests show 'inconclusive' for some series?

When the ratio limit L or root limit ρ equals 1 (common with p-series and rational functions), those specific tests cannot determine convergence, prompting the tool to evaluate p-series and integral tests.

### What is the starting index for the series?

The evaluation assumes the summation index n starts at n = 1.

### How is the total sum estimated for convergent series?

The sum is calculated by combining large-index partial sums with a tail estimate based on geometric or power-law decay.

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