Question

Let \( (a_n) \) be a sequence of positive real numbers such that \[ a_1 = 1, \quad a_{n+1} = 2a_n a_{n+1} - a_n = 0 \text{ for all } n \geq 1. \] Then the sum of the series \[ \sum_{n=1}^{\infty} a_n \] lies in the interval

• A. \( (1, 2) \)
• B. \( (2, 3) \)
• C. \( (3, 4) \)
• D. \( (4, 5) \)

Hint:

To solve recurrence relations, try to express the terms explicitly and look for patterns to find the sum of the series.

Answer 1

The Correct Option is A

Step 1: Understanding the sequence.
The recurrence relation for \( a_n \) is \( a_{n+1} = 2a_n + a_{n+1} - a_n \). We solve for \( a_n \), and find the sum of the series.

Step 2: Conclusion.
Thus, the sum of the series lies between \( 1 \) and \( 2 \), so the correct answer is \( \boxed{(A)} \).