Convergence of Series with $\cos(\pi n)$ and Denominator $(n+1)(n+2)$

Math Problem Statement

Solve

Does the series $\sum_{n=1}^{\infty} \frac{\cos(\pi n)}{(n+1)(n+2)}$ converge? Give reasons for your answer.

Solution

The given problem involves determining whether the series

$\sum_{n=1}^{\infty} \frac{\cos(\pi n)}{(n+1)(n+2)}$

converges, and to justify the answer.

Solution:

Step 1: Simplify the term $\cos(\pi n)$

The term $\cos(\pi n)$ alternates between $-1$ and $1$ , depending on whether $n$ is odd or even:

\begin{cases} 1 & \text{if } n \text{ is even}, \\ -1 & \text{if } n \text{ is odd}. \end{cases}$$ Thus, the series is **alternating**. --- #### Step 2: Check the general term's behavior The general term of the series is: $$a_n = \frac{\cos(\pi n)}{(n+1)(n+2)}.$$ Ignoring the oscillation caused by $$\cos(\pi n)$$, we focus on the denominator: $$(n+1)(n+2) = n^2 + 3n + 2.$$ For large $$n$$, the term $$n^2$$ dominates in the denominator, so the general term behaves like: $$a_n \sim \frac{\cos(\pi n)}{n^2}.$$ This suggests that the general term decays approximately as $$\frac{1}{n^2}$$, which is similar to a $$p$$-series with $$p = 2 > 1$$. Such series are known to converge. --- #### Step 3: Apply the Alternating Series Test The Alternating Series Test (Leibniz's criterion) states that an alternating series $$\sum (-1)^n b_n$$ converges if: 1. $$b_n \geq 0$$, 2. $$b_n$$ is monotonically decreasing, 3. $$\lim_{n \to \infty} b_n = 0$$. Here, the alternating part is due to $$\cos(\pi n)$$, and the magnitude of the term is: $$b_n = \frac{1}{(n+1)(n+2)}.$$ - $$b_n \geq 0$$ is clearly true (denominator is positive for all $$n$$). - $$b_n$$ is monotonically decreasing because the denominator $$(n+1)(n+2)$$ grows as $$n$$ increases. - $$\lim_{n \to \infty} b_n = \lim_{n \to \infty} \frac{1}{(n+1)(n+2)} = 0$$. Since all three conditions are satisfied, the series converges by the Alternating Series Test. --- #### Step 4: Conclude The series $$\sum_{n=1}^\infty \frac{\cos(\pi n)}{(n+1)(n+2)}$$ **converges** because: 1. The general term $$\frac{1}{(n+1)(n+2)}$$ decreases to $$0$$ as $$n \to \infty$$, 2. The series alternates in sign, 3. The Alternating Series Test is satisfied. --- Would you like me to explain any specific step further or explore related concepts? --- ### 5 Related Questions: 1. What is the $$p$$-series test, and why does it determine convergence? 2. Can the ratio test be applied to this series? Why or why not? 3. How does the behavior of $$\cos(\pi n)$$ affect the convergence of this series? 4. What happens if the numerator was $$1$$ instead of $$\cos(\pi n)$$? 5. What is the difference between absolute and conditional convergence? ### Tip: When analyzing alternating series, always verify if the series satisfies the Alternating Series Test before proceeding to other tests!

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Math Problem Analysis

Mathematical Concepts

Infinite Series
Alternating Series Test
p-Series

Formulas

$\cos(\pi n) = (-1)^n$
General term approximation: $a_n \sim \frac{1}{n^2}$

Theorems

Alternating Series Test
Convergence of p-Series

Suitable Grade Level

College or Advanced High School (Grades 11-12)

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