Use the Root Test to determine the convergence or divergence of the series. $$\sum _ { n = 1 } ^ { \infty } \left( \frac { 4 n } { 3 n + 1 } \right) ^ { n }$$
A) converges
B) diverges
C) Root Test inconclusive

Question

Quizplus · Accepted Answer

The Root Test involves taking the nth root of the absolute value of the nth term and examining the limit as n approaches infinity. For the given series, $\lim_{n \to \infty} \sqrt[n]{\left( \frac{4n}{3n+1} \right)^n} = \lim_{n \to \infty} \frac{4n}{3n+1}$. As $n \to \infty$, this limit approaches $\frac{4}{3}$, which is greater than 1. According to the Root Test, if this limit is greater than 1, the series diverges.

Use the Root Test to Determine the Convergence or Divergence ∑n=1∞(4n3n+1)n\sum _ { n = 1 } ^ { \infty } \left( \frac { 4 n } { 3 n + 1 } \right) ^ { n }n=1∑∞​(3n+14n​)n

Use the Root Test to Determine the Convergence or Divergence $\sum _ { n = 1 } ^ { \infty } \left( \frac { 4 n } { 3 n + 1 } \right) ^ { n }$