1. What is a Geometric Series?

A geometric series is the sum of the terms of a geometric sequence, where each term after the first is found by multiplying the previous term by a constant called the common ratio.

2. How Does the Calculator Work?

The calculator uses the geometric series formula:

Where:

• \( S \) — Sum of the series
• \( a \) — First term of the series
• \( r \) — Common ratio between terms
• \( n \) — Number of terms

Explanation: The formula calculates the sum of a finite geometric series by accounting for the first term, the common ratio between terms, and the total number of terms.

3. Importance of Geometric Series

Details: Geometric series are fundamental in mathematics and have applications in finance (compound interest), physics, computer science, and many other fields.

4. Using the Calculator

Tips: Enter the first term (a), common ratio (r), and number of terms (n). The common ratio cannot be 1 (for r=1, the sum is simply a×n).

5. Frequently Asked Questions (FAQ)

Q1: What if the common ratio is 1?
A: When r=1, the series becomes a simple arithmetic series where each term equals 'a', and the sum is a×n.

Q2: What about infinite geometric series?
A: For infinite series (n→∞), the sum converges to a/(1-r) only if |r| < 1.

Q3: Can the ratio be negative?
A: Yes, the ratio can be negative, resulting in an alternating series.

Q4: What are practical applications?
A: Used in calculating loan payments, investment growth, fractal geometry, and many physics applications.

Q5: How precise are the calculations?
A: The calculator provides results with 4 decimal places of precision.