Sum of Convergent Series Calculator

Geometric Series Formula:

\[ S = \frac{a}{1 - r} \quad \text{for} \quad |r| < 1 \]

Sum of Series:

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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:

\[ S = \frac{a}{1 - r} \quad \text{for} \quad |r| < 1 \]

Where:

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

Explanation: The series converges only when the absolute value of the common ratio is less than 1.

3. Importance of Convergence

Details: The convergence condition (|r| < 1) is crucial because it ensures the terms become smaller and approach zero, allowing for a finite sum.

4. Using the Calculator

Tips: Enter the first term (a) and common ratio (r). The calculator will only return a valid sum if |r| < 1.

5. Frequently Asked Questions (FAQ)

Q1: What happens if |r| ≥ 1?
A: The series diverges and doesn't have a finite sum. The calculator will show an error message.

Q2: Can I calculate partial sums with this calculator?
A: No, this calculator only computes the sum of an infinite geometric series when it converges.

Q3: What are some real-world applications?
A: Geometric series are used in finance (compound interest), physics (wave patterns), and computer science (algorithm analysis).

Q4: How accurate is the calculation?
A: The calculation is mathematically exact (within floating-point precision) for convergent series.

Q5: Can r be negative?
A: Yes, as long as its absolute value is less than 1. The series will alternate between positive and negative terms.