Teaching MPPSC Assistant Professor Mock Test Series 2025 Mathematical Science Analysis Sequences & Series (Convergence)
Let (xn) and (yn) be sequences of real numbers defined by \(x_{1}=1, \quad y_{1}=\frac{1}{2}, \quad x_{n+1}=\frac{x_{n}+y_{n}}{2}, \quad \text { and } \quad y_{n+1}=\sqrt{x_{n} y_{n}}\) for all n ∈ N. Then which one of the following is true?
1
(xn) is convergent, but (yn) is not convergent.
2
(xn) is not convergent, but (yn) is convergent.
3
Both (xn) and (yn) are convergent and \(\lim _{n \rightarrow \infty} x_{n}>\lim _{n \rightarrow \infty} y_{n}\)
4
Both (xn) and (yn) are convergent and \(\lim _{n \rightarrow \infty} x_{n}=\lim _{n \rightarrow \infty} y_{n}\)