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

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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