Direct Comparison Test Calculator

Direct Comparison Test Calculator calculator can be used to determine if an infinite series converges or diverges by comparing it to a known benchmark series.

Input Parameters

Series 1 (to be tested)

Enter the general term as a mathematical expression

Series 2 (benchmark)

Enter the general term as a mathematical expression

Calculation Results

Comparison Result

Direct Comparison Test Formula

If 0 ≤ bn ≤ an for all n and ∑bn converges/diverges, then ∑an also converges/diverges.

Where:
an = General term of the series to be tested
bn = General term of the benchmark series

Direct Comparison Test Calculator Calculator Usage Guide

Learn how to use the Direct Comparison Test Calculator and its working principles

What is the Direct Comparison Test?

The Direct Comparison Test is a method used in calculus to determine whether an infinite series converges or diverges by comparing it to a known benchmark series.

How to use this calculator:

  1. For Series 1 (to be tested), enter its general term (an). This is the series you want to analyze.
  2. Specify whether Series 1 is convergent (p-series with p>1) or divergent (p-series with p≤1).
  3. For Series 2 (benchmark), enter its general term (bn). This is a series with known convergence properties.
  4. Specify whether Series 2 is convergent or divergent.
  5. Click "Calculate" to see the comparison result.

Working Principle:

The Direct Comparison Test works by comparing the terms of the two series. If 0 ≤ bn ≤ an for all n, and if ∑bn converges, then ∑an also converges. Similarly, if 0 ≤ an ≤ bn for all n, and if ∑an diverges, then ∑bn also diverges.

Example:

Consider Series 1: an = 1/(n+1)² and Series 2: bn = 1/n². Since 1/n² ≤ 1/(n+1)² for all n, and ∑bn converges (as it's a p-series with p=2), then ∑an also converges.

Limitations:

The Direct Comparison Test can only be applied when you can find a second series with known convergence properties that bounds the first series. If one series is convergent and the other is divergent, you may need to use the Limit Comparison Test instead.