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The Divergence Test, also known as the nth-term test, is a simple yet powerful tool to determine the behavior of an infinite series. If you have a series \( \sum a_k \), the test states that if the limit of \( a_k \) as \( k \) approaches infinity is not zero, then the series diverges. This means the series doesn’t settle on a specific value; rather it spreads out indefinitely.
It is crucial to note, though, that if \( a_k \) approaches zero, the series is not guaranteed to converge. There could be other factors affecting convergence. The Divergence Test can confirm divergence, but not convergence. It's always a good first check because of its simplicity. In our example, the limit of \( \frac{k^2+1}{k^2+3} \) is 1, clearly not zero, leading directly to the conclusion that the series diverges.

In calculus and series analysis, understanding the limit of a sequence is foundational. A sequence is a list of numbers in a specific order, typically a function of another variable. When analyzing whether a series converges or diverges, looking at the limit of its sequence terms \( a_k \) as \( k \) approaches infinity can give vital insights.
To find the limit, you must simplify the expression to make its behavior more apparent. For polynomial fractions, dividing through by the highest power of \( k \) in both the numerator and denominator helps. In our series, by simplifying \( \frac{k^2+1}{k^2+3} \) to \( \frac{1 + \frac{1}{k^2}}{1 + \frac{3}{k^2}} \), it shows that as \( k \) becomes very large, the extra terms shrink affecting the limit minimally – leading to \( 1 \) in this case.

Polynomial fractions frequently appear in series analysis. They involve terms that are ratios of polynomials. Understanding these can simplify evaluating limits and determining series behavior. When comparing the polynomial degrees of the numerator and the denominator, patterns emerge that assist with simplification.
Key strategies include:

• Divide each term by the highest degree of \( k \) present.
• Observe the dominant terms when \( k \) approaches infinity.

These help reduce complex fractions into simpler forms, making them easier to evaluate at large \( k \). In our series, eliminating lower degree terms reveals the limit as \( 1 \), establishing divergence with the help of the Divergence Test.