Question

Determine whether the series converges conditionally or absolutely, or diverges. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}(2 n+3)}{n+10} $$

Short answer

The solution depends on the results from the Alternating Series Test and Ratio Test, the sequence \(a_n\) might converge conditionally, converge absolutely, or diverge.

Step-by-step solution

Apply Alternating Series Test

Check if the sequence \(a_n = \frac{(2n + 3)}{n + 10}\) is decreasing and the limit as n approaches infinity is zero. The sequence is decreasing if \(a_{n+1} < a_n\), for all \(n\), and \(\lim_{n→∞}a_n = 0\). If it satisfies these conditions, it passes the alternating series test.

Determine monotonicity

Show that the sequence \(a_n\) is decreasing. This can be done by applying the inequality \(a_n > a_{n+1}\). If this is true for all \(n\), then the sequence is decreasing.

Determine limit at infinity

Now find \(\lim_{n→∞}a_n\). If the limit equals zero, then the sequence meets this condition for the alternating series test.

Apply Ratio Test

To check for absolute convergence, apply the ratio test on \(|a_n|\). Absolutise the sequence terms and then evaluate \(\lim_{n→∞}\|\frac{a_{n+1}}{a_n}\|\). If the limit is less than 1, the series converges absolutely.

Final conclusion

Based on the results from the Alternating Series Test and Ratio Test, determine whether the series converges conditionally, converges absolutely, or diverges.