Suppose f_1(x) \sim g_1(x) and f_2(x) \sim g_2(x) as x \rightarrow \infty for some functions f_1,f_2,g_1,g_2. Then I know that f_1(x) + f_2(x) \sim g_1(x) + g_2(x) as x \rightarrow \infty, and that more generally we have \sum_{i=1}^k f_i(x) \sim \sum_{i=1}^k g_i(x) as x \rightarrow \infty for any fixed k \in \mathbb{N} if each f_i(x) \sim g_i(x).

But what if k is not fixed, and actually depends on x? For example, is it true that \sum_{0 \leq k \leq x} f_i(x) \sim \sum_{0 \leq k \leq x} g_i(x)?