Teaching MPPSC Assistant Professor Mock Test Series 2025 Mathematical Science Analysis Sequences & Series (Convergence)
Given (an)n≥1 a sequence of real numbers, which of the following statements is true?
1
\(\sum_{n ≥ 1}(-1)^n \frac{a_n}{1+\left|a_n\right|}\) converges
2
There is a subsequence \(\left(a_{n_k}\right)_{k \geq 1}\) such that \(\sum_{k ≥ 1} \frac{a_{n_k}}{1+\left|a_{n_k}\right|}\) converges
3
There is a number b such that \(\sum_{n \geq 1}\left|b-\frac{a_n}{1+\left|a_n\right|}\right|(-1)^n\) converges
4
There is a number b and a subsequence \(\left(a_{n_k}\right)_{k \geq 1}\) such that \(\sum_{k \geq 1}\left|b-\frac{a_{n_k}}{1+\left|a_{n_k}\right|}\right|\) converges