Question

Let \( s_n = 1 + \dfrac{(-1)^n}{n}, \, n \in \mathbb{N}. \) Then the sequence \(\{s_n\}\) is
 

• A. monotonically increasing and is convergent to 1
• B. monotonically decreasing and is convergent to 1
• C. neither monotonically increasing nor monotonically decreasing but is convergent to 1
• D. divergent

Hint:

A sequence involving \( (-1)^n \) often alternates and may not be monotonic, but it can still converge if the oscillations shrink to zero.

Answer 1

The Correct Option is C

Step 1: Write the given sequence. 
\( s_n = 1 + \dfrac{(-1)^n}{n} \). 
 

Step 2: Observe the behavior of the sequence. 
For even \( n \): \( s_n = 1 + \dfrac{1}{n} \) (greater than 1). 
For odd \( n \): \( s_n = 1 - \dfrac{1}{n} \) (less than 1). 
 

Step 3: Analyze monotonicity. 
The even and odd subsequences approach 1 from opposite sides, so the sequence alternates and is not monotonic. 
 

Step 4: Determine convergence. 
\(\lim_{n \to \infty} \dfrac{(-1)^n}{n} = 0 \Rightarrow \lim_{n \to \infty} s_n = 1.\) Thus, the sequence is convergent to 1. 
 

Final Answer: The sequence is not monotonic but convergent to 1.