Use the Ratio Test to determine the convergence or divergence of the series. $$\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 6 } } { 10 ^ { n } }$$
A) converges
B) diverges
C) Ratio Test inconclusive

Question

Quizplus · Accepted Answer

The Ratio Test involves taking the limit of the absolute value of the ratio of the (n+1)th term to the nth term as n approaches infinity. For the given series, $\lim _ { n \to \infty } \left| \frac { (n + 1) ^ { 6 } } { 10 ^ { n + 1 } } \cdot \frac { 10 ^ { n } } { n ^ { 6 } } \right| = \lim _ { n \to \infty } \frac { (n + 1) ^ { 6 } } { 10n ^ { 6 } }$. This simplifies to $\frac { 1 } { 10 } \lim _ { n \to \infty } \left( \frac { n + 1 } { n } \right) ^ { 6 }$, which equals $\frac { 1 } { 10 }$. Since this limit is less than 1, the series converges by the Ratio Test.

Use the Ratio Test to Determine the Convergence or Divergence ∑n=1∞n610n\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 6 } } { 10 ^ { n } }n=1∑∞​10nn6​

Use the Ratio Test to Determine the Convergence or Divergence $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 6 } } { 10 ^ { n } }$