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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 $n \to \infty$.

To determine if an infinite series converges or diverges when given an explicit formula for its $n$-th partial sum $S_n$, evaluate the limit $\lim_{n \to \infty} S_n$. If this limit exists and is a finite value, the series converges to that value; if the limit is infinite or does not exist, the series diverges.

Conditions: You are given an explicit algebraic formula for the $n$-th partial sum, $S_n$.; The limit is taken as $n \to \infty$.

To determine if an infinite series converges or diverges when given an explicit formula for its $n$-th partial sum $S_n$, evaluate the limit $\lim_{n \to \infty} S_n$. If this limit exists and is a finite value, the series converges to that value; if the limit is infinite or does not exist, the series diverges.

Conditions: You are given an explicit algebraic formula for the $n$-th partial sum, $S_n$.; The limit is taken 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 $n \to \infty$.

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 $n \to \infty$.

When evaluating the limit of a partial sum $S_n$ expressed as a quotient of polynomials in $n$ as $n \to \infty$, dividing both the numerator and denominator by the highest power of $n$ in the denominator (specifically $n^2$) isolates the asymptotic behavior of each term. This transformation converts lower-order terms into fractions with $n$ in the denominator, which clearly tend to 0, thereby revealing that the numerator dominates and the limit is infinity.

Conditions: The expression is a quotient of polynomials in $n$.; The limit is taken as $n \to \infty$.; The highest power of $n$ in the denominator is $n^2$.

When evaluating the limit of a partial sum $S_n$ expressed as a quotient of polynomials in $n$ as $n \to \infty$, dividing both the numerator and denominator by the highest power of $n$ in the denominator (specifically $n^2$) isolates the asymptotic behavior of each term. This transformation converts lower-order terms into fractions with $n$ in the denominator, which clearly tend to 0, thereby revealing that the numerator dominates and the limit is infinity.

Conditions: The expression is a quotient of polynomials in $n$.; The limit is taken as $n \to \infty$.; The highest power of $n$ in the denominator is $n^2$.