Show that the series $$\sum_{n=0}^{∞} \frac{1}{1 + n^2}$$ converges.

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Solution:

Since, f(n) = $$\frac{1}{1 + n^2}$$ which is continuous, positive and decreasing for n >= 0

Here,

$$\int_{0}^{∞} (\frac{1}{1 + x^2})dx$$

$$= \lim_{k \to ∞} \int_{0}^{k} \frac{dx}{1 + x^2}$$

$$= \lim_{k \to ∞} [tan^{-1}x]_{0}^{k}$$

$$= \lim_{k \to ∞} [tan^{-1}(k) – tan^{-1}(0)]$$

$$= \frac{\pi}{2}$$

This means the integral converges. Therefore, by integral test, the given test is convergent.