Convergence Calculator

The Convergence Calculator enables users to calculate the sum of a geometric series for a given initial value, common ratio, number of terms, and desired precision, while also providing insights on convergence rate, last term, infinite sum, and error estimation.

Use Our Convergence Calculator

How to Use the Convergence Calculator

The Convergence Calculator is a tool designed to help you compute various attributes of a geometric series, including the sum of the first n terms, sum to infinity, rate of convergence, last term value, and error estimate. Follow these steps to correctly utilize the calculator:

Step 1: Enter Initial Value

Begin by entering the initial value of the geometric sequence.

  • Find the input field labeled as Initial Value.
  • Input a number within the range of -1,000,000 to 1,000,000.
  • This field is required, so ensure you make a valid input to proceed.

Step 2: Enter the Common Ratio

Next, specify the common ratio of the sequence. This ratio determines how the terms progress.

  • Locate the input field labeled Common Ratio (r).
  • Enter a decimal value such that its absolute value is less than 1 (|r| < 1).
  • Again, this field is mandatory, and entering a valid ratio is crucial for further calculations.

Step 3: Enter the Number of Terms

Indicate the number of terms you want to consider in the sequence.

  • Find the field labeled Number of Terms.
  • Input an integer value between 1 and 1,000, as it dictates up to which term the calculations apply.

Step 4: Choose Decimal Precision

Select your desired precision level for the answers.

  • Look for the Decimal Precision dropdown menu.
  • Choose from the available options: 2, 4, 6, or 8 decimal places.
  • This choice affects the precision level of all numerical results presented by the calculator.

Step 5: Calculate and Review Results

Once all inputs are provided, the calculator will automatically compute the results. Here’s what you can expect to see:

  • Sum of First n Terms: Calculated as initialValue * (1 – pow(ratio, terms)) / (1 – ratio).
  • Sum to Infinity: Indicates the sum of the sequence to infinity if |r| < 1; otherwise, it shows ‘Divergent’.
  • Rate of Convergence: Displays as the absolute value of the common ratio.
  • Last Term Value: Calculated as initialValue * pow(ratio, terms – 1).
  • Error Estimate: Shows the error estimate if |r| < 1; otherwise, ‘N/A’.

By following these steps, you can effectively use the Convergence Calculator to analyze geometric sequences and get precise results.