$Is $\sum_{i=1}^{\infty}\arctan (\frac{n+1}{n^2+5})$ convergent or divergent?$

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$Is $\sum_{i=1}^{\infty}\arctan (\frac{n+1}{n^2+5})$ convergent or divergent?$

Is the infinite series
$$\sum_{i=1}^{\infty}\arctan (\frac{n+1}{n^2+5})$$ convergent or divergent?

Calculus Sequences and Series Trigonometric Functions

Sophiam

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Since $\lim_{x \rightarrow 0}\frac{\arctan x}{x}=1$, we have
\[\lim_{n \rightarrow \infty} \frac{\arctan{\frac{n+1}{n^2+5}{}}}{\frac{1}{n}}=\lim_{n \rightarrow \infty} \frac{\arctan{\frac{n+1}{n^2+5}{}}}{\frac{n+1}{n^2+5}}\times \frac{\frac{n+1}{n^2+5}}{\frac{1}{n}}\]
\[ =\lim_{n \rightarrow \infty} \frac{\frac{n+1}{n^2+5}}{\frac{1}{n}}=1. \]
By limit comparision theorem, since $\sum_{i=1}^{\infty}\frac{1}{n}$ diverges, the infinite series $$\sum_{i=1}^{\infty}\arctan \frac{n+1}{n^2+5}$$ also diverges.

Erdos

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$Is $\sum_{i=1}^{\infty}\arctan (\frac{n+1}{n^2+5})$ convergent or divergent?$

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