Why an Infinite Limit of Partial Sums Implies Divergence

Conditions

• The series is represented as $S = \lim_{n\to\infty} S_n$.
• The limit of the partial sums $S_n$ is evaluated as $n \to \infty$.

Reasoning, step by step

1. Recall the definition of an infinite series as the limit of its partial sums.
2. Evaluate the limit of the partial sums.
3. Check if the limit is a finite value.
4. Conclude that an infinite limit means the series does not converge to a finite value, hence it diverges.

Example

Since $\lim_{n\to\infty} S_n = \infty$, the series does not approach a finite value. The clip states the criterion explicitly: convergence would require the partial sums to approach some finite limit.

Common misconceptions

• Thinking that a series diverges only if its individual terms do not approach zero.
• Confusing the divergence of the sequence of partial sums with the divergence of the individual terms.

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