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An infinite series $S$ is defined as the limit of its partial sums $S_n$ as $n \to \infty$. For the series to converge, this limit must be a finite value. If $\lim_{n\to\infty} S_n = \infty$, the sum grows without bound and does not approach a finite number, so the series diverges.

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

An infinite series $S$ is defined as the limit of its partial sums $S_n$ as $n \to \infty$. For the series to converge, this limit must be a finite value. If $\lim_{n\to\infty} S_n = \infty$, the sum grows without bound and does not approach a finite number, so the series diverges.

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