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Say I have a series $\sum_{k=1}^n f(k)$ for some function $f$. I want to find the biggest $n \in \mathbb{N}$ such that $\sum_{k=1}^n f(k) \leq c$ for some $c \in \mathbb{R}$ without actually computing the partial sums. Is there a method for finding such $n$ for general $f$?

For example take

$$\sum_{k=1}^n \binom{12}{k} \frac{1}{k} \leq 400.$$

If I am not mistaken, this should hold for $n \leq 4$. How would I find this without computing the partial sums? Thanks in advance for any help.

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