Question

determine whether the series is convergent or divergent. if it is convergent, find its sum. (if the quantity diverges, enter diverges.)
$$ sum _ { n = 1 } ^ { infty } arctan ( 2 n ) $$

Explanation:

Step1: Apply the Divergence Test

The Divergence Test states that for a series \(\sum_{n = 1}^{\infty}a_{n}\), if \(\lim_{n
ightarrow\infty}a_{n}
eq0\), then the series \(\sum_{n = 1}^{\infty}a_{n}\) diverges.
For the series \(\sum_{n = 1}^{\infty}\arctan(2n)\), we find \(\lim_{n
ightarrow\infty}\arctan(2n)\).
We know that the function \(y = \arctan(x)\) has the property \(\lim_{x
ightarrow\infty}\arctan(x)=\frac{\pi}{2}\).
Let \(x = 2n\), as \(n
ightarrow\infty\), \(x = 2n
ightarrow\infty\). So, \(\lim_{n
ightarrow\infty}\arctan(2n)=\frac{\pi}{2}
eq0\).

Answer:

DIVERGES