Determine whether the series \(\sum_{n=1}^∞ \frac{n^2}{5n^2 + 4}\) converges or diverges.

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Solution:

Given

\(\sum_{n=1}^∞ \frac{n^2}{5n^2 + 4}\)

Here

\(a_n =  \frac{n^2}{5n^2 + 4}\)

Now,

\( = \lim_{n \to ∞} \frac{n^2}{5n^2 + 4}\)

\( = \lim_{n \to ∞} \frac{1}{5 + \frac{4}{n^2}}\)

\(= \frac{1}{5+0}\)

\(= \frac{1}{5} \neq 0\)

Therefore, by nth term test for divergence, the given series is divergent.

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