By calculating an appropriate number of terms, determine if the series converges or diverges. If it converges, find the limit $L$ and the smallest integer $N$ such that $\left| a _ { n } - L \right| < 0.01$ for $n \geq N$ ; otherwise indicate divergence.
- $a _ { n } = \frac { \tan ^ { - 1 } n } { e ^ { n } }$

Question

By calculating an appropriate number of terms, determine if the series converges or diverges. If it converges, find the limit $L$ and the smallest integer $N$ such that $\left| a _ { n } - L \right| < 0.01$ for $n \geq N$; otherwise indicate divergence.
-$$a _ { n } = \frac { \tan ^ { - 1 } n } { e ^ { n } }$$
A) $\mathrm { L } = \frac { \pi } { \mathrm { e } } , \mathrm { N } = 8$
B) $\mathrm { L } = \frac { \pi } { 2 \mathrm { e } } , \mathrm { N } = 6$
C) $L = 0 , N = 5$
D) diverges

Quizplus · Accepted Answer

The series converges to $L = 0$ because $\tan^{-1}n$ approaches $\frac{\pi}{2}$ as $n$ increases, but the denominator $e^n$ grows much faster, making the fraction approach 0. The specific $N$ value cannot be determined without explicit calculations, but the logic for convergence is correct.

By Calculating an Appropriate Number of Terms, Determine If the Series