Question

If \( s_n = \dfrac{(-1)^n}{2^n + 3} \) and \( t_n = \dfrac{(-1)^n}{4n - 1}, \, n = 0, 1, 2, \dots, \) then
 

• A. \( \sum_{n=0}^\infty s_n \) is absolutely convergent
• B. \( \sum_{n=0}^\infty t_n \) is absolutely convergent
• C. \( \sum_{n=0}^\infty s_n \) is conditionally convergent
• D. \( \sum_{n=0}^\infty t_n \) is conditionally convergent

Hint:

Use the Alternating Series Test when terms alternate in sign and decrease to zero; check absolute convergence separately.

Answer 1

The Correct Option is A, D

Step 1: Check absolute convergence of \( s_n. \) 
\[ |s_n| = \frac{1}{2^n + 3}. \] This behaves like \( \frac{1}{2^n} \), which forms a convergent geometric series. Hence, \( \sum |s_n| \) converges \( \Rightarrow \sum s_n \) is absolutely convergent.

Step 2: Check absolute convergence of \( t_n. \) 
\[ |t_n| = \frac{1}{4n - 1}. \] This behaves like \( \frac{1}{n} \), which diverges. Hence, \( \sum |t_n| \) diverges, but \( \sum t_n \) is an alternating series with decreasing terms tending to zero. By the Alternating Series Test, it converges conditionally.

Final Answer: \( s_n \) is absolutely convergent and \( t_n \) is conditionally convergent. \[ \boxed{\text{(A) and (D)}} \]