Use the Direct Comparison Test to determine the convergence or divergence of the series $\sum _ { n = 0 } ^ { \infty } e ^ { - n ^ { 6 } }$.
A)The series $\sum _ { n = 0 } ^ { \infty } e ^ { - n ^ { 6 } }$ diverges.
B) The series $\sum _ { n = 0 } ^ { \infty } e ^ { - n ^ { 6 } }$ converges.

Question

Quizplus · Accepted Answer

We can use the Direct Comparison Test with the series $\sum _ { n = 0 } ^ { \infty } \frac{1}{n ^ { 6 }}$, which converges (since it is a p-series with p = 6 > 1). Note that for all positive integers n, we have $e^{-n^6} < \frac{1}{n^6}$ (this can be shown by taking the logarithm of both sides and using the fact that $\ln x < x$ for all x > 0). Therefore, by the Direct Comparison Test, the series $\sum _ { n = 0 } ^ { \infty } e ^ { - n ^ { 6 } }$ converges.

Use the Direct Comparison Test to Determine the Convergence or Divergence