Use the Root Test to determine the convergence or divergence of the series. $$\sum _ { n = 1 } ^ { \infty } \left( \frac { 7 n + 1 } { 4 n - 1 } \right) ^ { n }$$
A) converges
B) diverges
C) Root Test inconclusive

Question

Quizplus · Accepted Answer

Applying the root test, we have
$$
\lim_{n	o\infty}\sqrt[n]{\left|\left(\frac{7n+1}{4n-1}ight)^night|}=\lim_{n	o\infty}\frac{7n+1}{4n-1}.
$$
This limit is equal to $7/4>1$, so by the root test, the series diverges. Therefore, the answer is (B) diverges.

Use the Root Test to Determine the Convergence or Divergence ∑n=1∞(7n+14n−1)n\sum _ { n = 1 } ^ { \infty } \left( \frac { 7 n + 1 } { 4 n - 1 } \right) ^ { n }n=1∑∞​(4n−17n+1​)n

Use the Root Test to Determine the Convergence or Divergence $\sum _ { n = 1 } ^ { \infty } \left( \frac { 7 n + 1 } { 4 n - 1 } \right) ^ { n }$