Let $(c_n)_{n \in \mathbf{N}^*}$ and $(d_n)_{n \in \mathbf{N}^*}$ be two sequences of strictly positive real numbers such that: $c_n \underset{n \to +\infty}{\sim} d_n$ and the series $\sum_n c_n$ diverges. We admit without proof the following result: Theorem 1. Let $(a_n)_{n \in \mathbf{N}^*}$ and $(b_n)_{n \in \mathbf{N}^*}$ be two sequences of nonzero real numbers such that $a_n = o(b_n)$ as $n \to +\infty$ and the series $\sum_n |b_n|$ is divergent. Then: $$\sum_{k=1}^n a_k = o\!\left(\sum_{k=1}^n |b_k|\right) \text{ as } n \to +\infty.$$ By using this result, show that the series $\sum_n d_n$ is divergent and that: $$\sum_{k=1}^n c_k \underset{n \rightarrow +\infty}{\sim} \sum_{k=1}^n d_k.$$
Let $(c_n)_{n \in \mathbf{N}^*}$ and $(d_n)_{n \in \mathbf{N}^*}$ be two sequences of strictly positive real numbers such that: $c_n \underset{n \to +\infty}{\sim} d_n$ and the series $\sum_n c_n$ diverges.
We admit without proof the following result:
\textbf{Theorem 1.} Let $(a_n)_{n \in \mathbf{N}^*}$ and $(b_n)_{n \in \mathbf{N}^*}$ be two sequences of nonzero real numbers such that $a_n = o(b_n)$ as $n \to +\infty$ and the series $\sum_n |b_n|$ is divergent. Then:
$$\sum_{k=1}^n a_k = o\!\left(\sum_{k=1}^n |b_k|\right) \text{ as } n \to +\infty.$$
By using this result, show that the series $\sum_n d_n$ is divergent and that:
$$\sum_{k=1}^n c_k \underset{n \rightarrow +\infty}{\sim} \sum_{k=1}^n d_k.$$