If $\sum\limits_{n=2}^∞\frac1{f(n)}=ζ(C)-1$, is it ok to write $\lim\limits_{n→∞}f(n)=n^C$?

Question

Consider a function $f(n)$ such that $$\sum_{n=2}^\infty \frac{1}{f(n)} = \zeta (C)-1 = \sum_{n=2}^\infty \frac{1}{n^C}.$$ Is it ok to write $$\lim_{n\to \infty} f(n)=n^C?$$