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The partial sum $S_n$ is the finite sum of the first $n$ terms, whereas the infinite series $S$ is defined as the limit of $S_n$ as $n \to \infty$. While $S_n$ forms a sequence of values, $S$ represents the single limiting value (or divergence) that this sequence approaches.

Conditions: $S_n$ denotes the sum of the first $n$ terms.; $S$ denotes the infinite series $\sum_{n=1}^{\infty} a_n$.; The limit is taken as $n \to \infty$.

The partial sum $S_n$ is the finite sum of the first $n$ terms, whereas the infinite series $S$ is defined as the limit of $S_n$ as $n \to \infty$. While $S_n$ forms a sequence of values, $S$ represents the single limiting value (or divergence) that this sequence approaches.

Conditions: $S_n$ denotes the sum of the first $n$ terms.; $S$ denotes the infinite series $\sum_{n=1}^{\infty} a_n$.; The limit is taken as $n \to \infty$.