How to prove this result with sequences and series? Let $\left(v_n\right)_{n\geqslant1}$ and $\left(w_n\right)_{n\geqslant1}$ be two real sequences with strictly positive values, i.e., $v_n> 0$ and $w_n >0$. Assume that $v_n\sim w_n$ and the series $\sum_k v_k$ diverges.
Hence: $\sum_kv_k\sim\sum_kw_k.$ Which theorem is this? How can we prove it?

Hint: Since $\frac{v_n}{w_n}\rightarrow 1$ as $n\rightarrow\infty$, there is some $N$ such that for all $n>N$, $\frac{v_n}{w_n}<2$.  Then, for all $n>N$, $v_n<2w_n$.  Therefore, $\sum v_n<C+2\sum w_n$.  Therefore, if $\sum w_n$ converges, then so does $\sum v_n$.  You can use a similar argument replacing $2$ with $\frac{1}{2}$.
