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Understanding whether an infinite series converges or diverges is critical in mathematics. One tool employed to make this determination is the Divergence Test. This test provides a straightforward initial check. It states: "If the limit of the sequence of terms \( a_n \) does not approach zero as \( n \) approaches infinity, then the series \( \sum_{n=1}^{\infty} a_n \) must diverge." This doesn't necessarily mean that if the limit is zero, the series converges; other tests may be needed for such a case. The Divergence Test is thus invaluable when confronted with non-zero limits, promptly concluding divergence without further analysis.

An infinite series is the sum of infinitely many terms. It is represented by the notation \( \sum_{n=1}^{\infty} a_n \). Understanding these series lies at the heart of calculus and analysis, providing pathways for interpreting complex functions. Infinite series have various types, such as geometric, harmonic, and alternating series, each with rules determining convergence or divergence. When we examine an infinite series, we usually want to find out whether it converges (approaches a finite value) or diverges (increases without bound). In our example, knowing \( \frac{n^2}{n^2+1} \) does not shrink to zero implies divergence, overriding other considerations.

The concept of the limit of a sequence involves understanding what value, if any, the terms of a sequence approach as they progress towards infinity. Mathematically, when we say the limit of a sequence \( a_n \) is \( L \), denoted by \( \lim_{n \to \infty} a_n = L \), it means that as \( n \) becomes very large, \( a_n \) gets arbitrarily close to \( L \). In practice, this helps us analyze how terms behave in infinite series. In our series, since \( \lim_{n \to \infty} \frac{n^2}{n^2+1} = 1 \) (not zero), it signals to us via the divergence test that the series does not settle to a specific value. This illustrates the significance of evaluating limits when assessing series behavior.