Why is a finite plot of partial sums not a proof of convergence?

Conditions

• The plot shows only a finite number of partial sums.
• Convergence is defined as the limit of the infinite sequence of partial sums.

Reasoning, step by step

1. Recognize that a finite plot provides empirical evidence, not a mathematical proof.
2. Understand that convergence depends on the behavior of the series as $n$ approaches infinity.
3. Apply a convergence test (like the alternating-series test) that analyzes the infinite properties of the terms.
4. Conclude that without such a test, a finite plot is insufficient to prove convergence.

Example

The video shows an animation of partial sums oscillating closer to $\ln(2)$, but explicitly states that this visual alone is not a proof of convergence.

Common misconceptions

• Believing that visual patterns in finite plots guarantee infinite behavior.
• Confusing empirical observation with mathematical proof.

Watch the explanation

Connected concepts

Explore next

• What constitutes a rigorous proof of convergence for an alternating series?
• How does the alternating-series test address the limitations of finite plots?