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

Answer: (C)

$T = 2S$ and $S \neq 0$.

Answer 2

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