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.
- $a _ { n } = \sqrt [ n ] { n + 1 }$

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.
-$$a _ { n } = \sqrt [ n ] { n + 1 }$$
A) L = ln 2, N = 394
B) L = 1, N = 394
C) L = 1, N = 652
D) diverges

Quizplus · Accepted Answer

The sequence $a_n = \sqrt[n]{n+1}$ converges to 1 as $n$ approaches infinity. Calculating further, the smallest integer $N$ such that $|a_n - 1| < 0.01$ is 652.

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