Series Convergence Calculator

The Series Convergence Calculator helps users determine whether a geometric, p-series, or harmonic series converges or diverges, calculate its partial sum, and if applicable, find the theoretical sum and rate of convergence.

Use Our Series Convergence Calculator

How to Use the Series Convergence Calculator

This guide will walk you through the steps to effectively use the Series Convergence Calculator for determining the convergence of series and calculating their sums.

Step 1: Select the Series Type

  • Locate the Select Series Type dropdown menu.
  • Select the type of series you want to analyze. The available options are:
    • Geometric Series: Use this option if your series is of the geometric type.
    • P-Series: Use this option for P-series.
    • Harmonic Series: Select this if your series has harmonic characteristics.

Step 2: Enter the First Term

  • In the First Term (a₁) input field, enter the first term of your series.
  • The value should be between -1,000,000 and 1,000,000.

Step 3: Enter Additional Parameters

The additional parameters depend on the type of series selected:

  • Geometric Series
    • Enter the Common Ratio (r) in its respective input field.
    • The ratio should be between -100 and 100, with increments of 0.01.
  • P-Series
    • Enter the Exponent (p) in its respective input field.
    • The exponent should be between 0 and 100, with increments of 0.1.

Step 4: Define the Number of Terms

  • Input the number of terms you wish to consider in the Number of Terms to Calculate field.
  • This value should range from 1 to 1,000.

Step 5: Review Results

Once all inputs are provided, the calculator computes the results automatically:

  • Convergence Status
    • For a Geometric Series: It shows “Converges” if the absolute ratio is less than 1. Otherwise, it shows “Diverges”.
    • For a P-Series: It shows “Converges” if the exponent is greater than 1. Otherwise, it shows “Diverges”.
    • For a Harmonic Series: It always shows “Diverges”.
  • Partial Sum
    • Displays the calculated sum of the specified number of terms in the series.
  • Theoretical Sum
    • For a convergent Geometric Series: Displays the theoretical infinite sum.
    • Otherwise, it is marked as “Undefined”.
  • Rate of Convergence
    • For a convergent Geometric Series: Displays the absolute value of the ratio.
    • For a convergent P-Series: Displays the reciprocal of the exponent.
    • Otherwise, it shows “Not Applicable”.