Sequence Convergence Calculator

What Is Sequence Convergence?

A sequence converges if its terms approach a specific number, called the limit, as the index \( n \) becomes very large. If no such number exists, the sequence diverges.

Formal Definition

A sequence \( a_n \) converges to \( L \) if for every \( \epsilon > 0 \), there exists \( N \in \mathbb{N} \) such that for all \( n > N \):
\( |a_n - L| < \epsilon \)

• a_n: the general term of the sequence
• L: the limit value
• N: an index beyond which all terms stay within ε of L

Why It Matters

• 📈 Understand long-term behavior of functions and algorithms
• 🔢 Foundation for series and integral tests in calculus
• 🧮 Helps in modeling physics, economics, and engineering systems

Real-Life Analogy

Imagine you're filling a glass drop by drop. If each drop brings the water level closer to the brim without ever spilling, you're converging. But if the level keeps jumping or never settles, the process diverges.

How to Use the Sequence Convergence Calculator

1. Enter the formula for \( a_n \), such as 1/n or (-1)^n / n
2. Click "Calculate" to analyze the limit as \( n \to \infty \)
3. View whether the sequence converges, diverges, or oscillates

Tip: This tool supports rational, exponential, alternating, and trigonometric sequences.

Sequence Convergence Examples

• Example 1: \( a_n = \frac{1}{n} \) → Converges to 0
• Example 2: \( a_n = \frac{(-1)^n}{n} \) → Converges to 0
• Example 3: \( a_n = (-1)^n \) → Diverges (oscillates)
• Example 4: \( a_n = \frac{n}{n+1} \) → Converges to 1
• Example 5: \( a_n = \ln(n) \) → Diverges to ∞
• Example 6: \( a_n = e^{-n} \) → Converges to 0

Applications of Sequence Convergence

• 📐 Limit proofs in real analysis
• 🧠 Evaluating series and infinite sums
• 💻 Stability in numerical methods
• 📊 Economic forecasting and signal processing

Frequently Asked Questions (FAQ)