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When the degree of the numerator is strictly greater than the degree of the denominator in a rational function, the function grows without bound as the variable approaches infinity. Consequently, the limit is infinity.

Conditions: The function is a quotient of two polynomials.; The degree of the numerator is greater than the degree of the denominator.; The variable approaches infinity.

When the degree of the numerator is strictly greater than the degree of the denominator in a rational function, the function grows without bound as the variable approaches infinity. Consequently, the limit is infinity.

Conditions: The function is a quotient of two polynomials.; The degree of the numerator is greater than the degree of the denominator.; The variable approaches infinity.

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 $n \to \infty$.

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 $n^2$.