Question

Let \( T \) denote the sum of the convergent series
\[ 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \ldots + (-1)^{n+1} \frac{1}{n} + \ldots\]
and let \( S \) denote the sum of the convergent series
\[ 1 - \frac{1}{2} - \frac{1}{4} + \frac{1}{3} - \frac{1}{6} - \frac{1}{8} + \frac{1}{5} - \frac{1}{10} - \frac{1}{12} + \sum_{n=1}^{\infty} a_n\]
where
\[ a_{3m-2} = \frac{1}{2m-1} , a_{3m-1} = 0, \text{ and } a_{3m} = \frac{-1}{4m} \text{ for } m \in \mathbb{N}.\]
Then which one of the following is true?

• A. \( T = S \) and \( S \neq 0 \).
• B. \( 2T = S \) and \( S \neq 0 \).
• C. \( T = 2S \) and \( S \neq 0 \).
• D. \( T = S = 0 \).

Answer 1

The Correct Option is C

The correct option is (C): \( T = 2S \) and \( S \neq 0 \).