For each of the following sequences, if the divergence test...

For each of the following sequences, if the divergence test applies, either state that $$\lim _{n \rightarrow \infty} a_{n}$$ does not exist or find $$\lim _{n \rightarrow \infty} a_{n}$$. If the divergence test does not apply, state why.

$$a_{n}=\frac{n}{5 n^{2}-3}$$

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Key Concepts

Divergence Test

The divergence test, also known as the nth term test for divergence, is a basic tool in the analysis of series. This test states that if the limit of the sequence a? as n approaches infinity is not zero (or does not exist), then the series ? a? must diverge. However, if the limit is zero, the test is inconclusive and does not guarantee convergence of the series.

Limit Calculation for Rational Functions

When computing limits of sequences defined by rational functions (quotients of polynomials), one effective method is to divide the numerator and denominator by the highest power of n present in the denominator. This approach leverages the dominance of the highest-degree term to simplify the limit, since lower-degree terms become negligible as n approaches infinity.

Sequence Convergence

Sequence convergence concerns whether a sequence approaches a specific value as n becomes infinitely large. In many cases, finding the limit of a sequence is essential for determining the behavior of associated series, and in the context of tests like the divergence test, it forms a key criterion for assessing convergence or divergence.

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