Problem 34 - Essential Calculus

Question

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

Step-by-Step Solution

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 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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.

Step 5: Final conclusion

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

Key Concepts

Alternating Series Test

The Alternating Series Test is a valuable tool when analyzing series that cycle signs, such as positive and negative terms. An alternating series takes the form \( \sum (-1)^n a_n \), where the \( a_n \) are positive terms. The series converges if two conditions are satisfied:

• The sequence \( a_n \) is decreasing.
• The limit of \( a_n \) as \( n \) approaches infinity is zero (\( \lim_{{n \to \infty}} a_n = 0 \)).

To apply the test, we first examine whether \( a_n = \frac{2n+3}{n+10} \) decreases. We compare terms: \( a_{n+1} \) and \( a_n \). If \( a_{n+1} < a_n \) for all \( n \), the sequence is decreasing. Then, calculate \( \lim_{{n \to \infty}} a_n \). If it tends to zero, the alternating series converges. This test, however, does not reveal if the convergence is absolute or conditional.

Ratio Test

The Ratio Test helps determine absolute convergence of a series by examining the limit of the ratio of successive terms. For a series \( \sum a_n \), apply the test by evaluating \( \lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right| \).

• If the resulting limit is less than 1, the series converges absolutely.
• If the limit is greater than 1, or infinite, the series diverges.
• If the limit equals 1, the test is inconclusive.

For our series, we simplify the expression \( \frac{a_{n+1}}{a_n} = \frac{(2(n+1)+3)/(n+11)}{(2n+3)/(n+10)} \). We then take the limit to investigate. If \( \left| \lim_{{n \to \infty}} \frac{a_{n+1}}{a_n} \right| < 1 \), this confirms absolute convergence.

Conditional Convergence

Conditional convergence occurs when a series \( \sum a_n \) converges but does not converge absolutely. Relative to convergence tests, a series may pass the Alternating Series Test (confirming its convergence) but fail the Ratio Test (indicating lack of absolute convergence).

• The series converges conditionally when it satisfies criteria to converge as an alternating series, but not via absolute methods.
• This typically implies the values at large terms cancel each other out.

For our series, if the Alternating Series Test is passed but the Ratio Test shows \( \lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right| \geq 1 \), then it converges conditionally. This hints at a delicate balance between positive and negative contributions leading to overall convergence.

Absolute Convergence

Absolute convergence is considered stronger than conditional convergence. It implies that a series \( \sum a_n \) converges even when all terms are considered positive, i.e., when using their absolute values. For a series \sum a_n to converge absolutely:

• The altered series \sum |a_n| must converge.
• Tests such as the Ratio Test can demonstrate its convergence.

When applied to our series, we check if the conditions hold when focusing solely on the magnitude of terms. Passing the Ratio Test suggests the series \( \sum |a_n| \) converges, establishing that the original series converges in this stronger, absolute sense, without depending on sign alterations between terms.