Direct Comparison Test $$ \sum_{n=1}^{\infty} \frac{1}{n^{6}+5} $$

Question

UpStudy AI · Accepted Answer

The series $$ \sum_{n=1}^{\infty} \frac{1}{n^{6}+5} $$ converges because it is less than the convergent $$p$$-series $$ \sum_{n=1}^{\infty} \frac{1}{n^{6}} $$. To apply the Direct Comparison Test to the series $$ \sum_{n=1}^{\infty} \frac{1}{n^{6}+5}, $$ we need to find a suitable comparison series. A good candidate is the series $$ \sum_{n=1}^{\infty} \frac{1}{n^{6}}. $$ We know that the series $$\sum_{n=1}^{\infty} \frac{1}{n^{6}}$$ converges because it is a $$p$$-series with $$p = 6 > 1$$. Next, we will compare the terms of our series with the terms of the comparison series. For $$n \geq 1$$, we have: $$ n^6 + 5 > n^6. $$ This implies: $$ \frac{1}{n^6 + 5} < \frac{1}{n^6}. $$ Thus, we can write: $$ 0 < \frac{1}{n^6 + 5} < \frac{1}{n^6} \quad \text{for all } n \geq 1. $$ Since $$\sum_{n=1}^{\infty} \frac{1}{n^6}$$ converges, by the Direct Comparison Test, we conclude that $$ \sum_{n=1}^{\infty} \frac{1}{n^{6}+5} $$ also converges. Therefore, the series $$ \sum_{n=1}^{\infty} \frac{1}{n^{6}+5} $$ converges.

Direct Comparison Test \( \sum_{n=1}^{\infty} \frac{1}{n^{6}+5} \)

UpStudy ThothAI Solution

Step-by-step Solution

Quick Answer

Direct Comparison Test \( \sum_{n=1}^{\infty} \frac{1}{n^{6}+5} \)

UpStudy ThothAI Solution

Step-by-step Solution

Quick Answer

Related Questions