​$$\text{Given } (a_n)_{n \geq 1} \text{ a sequence of real numbers}\\ \text{, which of the following statements is true?}$$​​

Correct option is D

​$$\sum_{n=1}^\infty (-1)^n \frac{a_n}{1 + |a_n|}\\\text{(i) Let } a_n = n.\text{ Then:}\sum_{n=1}^\infty (-1)^n \frac{n}{1 + n} \text{ does not converge.}\\\text{So, Option A is incorrect.}\\\text{Now, let } a_{n_k} \text{ be a subsequence.}\\ \text{If } a_n \text{ is convergent, then:}\lim_{n \to \infty} a_n = \lim_{n \to \infty} a_{n_k}.\\\text{This implies:}\lim_{n \to \infty} \frac{a_{n_k}}{1 + |a_{n_k}|} \text{ exists.}\\\text{If } a_n \text{ diverges to } \pm\infty \text{ or oscillates, we can still find a number such that:}\\ \lim_{n \to \infty} a_{n_k} = l \quad \text{or} \quad \lim_{n \to \infty} a_{n_k} = \pm\infty.\\\text{In both cases:}\lim_{n \to \infty} \frac{a_{n_k}}{1 + |a_{n_k}|} \text{ exists.}\\\text{Thus, we can say that for every sequence } (a_n), \\ \text{ there exists a number } b \text{ such that:}\\\sum_{k=1}^\infty \left| b - \frac{a_{n_k}}{1 + |a_{n_k}|} \right| \text{ converges.}\\\textbf{So, Option D is correct.}$$​​