Teaching MPPSC Assistant Professor Mock Test Series 2025 Mathematical Science Analysis Sequences & Series (Convergence)
Let T denote the sum of the convergent series \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\cdots+\frac{(-1)^{n+1}}{n}+\cdots\) and let S denote the sum of the convergent series \(1-\frac{1}{2}-\frac{1}{4}+\frac{1}{3}-\frac{1}{6}-\frac{1}{8}+\frac{1}{5}-\frac{1}{10}-\frac{1}{12}+\cdots=\sum_{n=1}^{\infty} a_{n}\) where\(a_{3 m-2}=\frac{1}{2 m-1}, \:\:a_{3 m-1}=\frac{-1}{4 m-2} \:\:\text { and } \:\: a_{3 m}=\frac{-1}{4 m}\) for m ∈ N. Then which one of the following is true?
1
T = S and S ≠ 0.
2
2T = S and S ≠ 0.
3
T = 2S and S ≠ 0.
4
T = S = 0.