First of all, Happy new year!
Sorry for the non specific title, There was not enough space for a good description.
I have been trying to find a example for the following function $f$:
Does there exists a uniformly continuous function $f$ at $[-∞,+∞] $Such that the series $\sum_{n=1}^{\infty} \frac{f(n)}{n^3}$ will diverge?
Assuming theres such function $f$ as above, Can we find another function such that the Series $\sum_{n=1}^{\infty} \frac{f(n)}{2^n}$ will diverge?
I am positive theres no such function $f$ for the second question, But im not sure how to disprove this statement, And I was not able to find such $f$ for my first question, Which im not even sure If such function exists.
The fastest growing uniformly continous $f$ I can think of is $f(x) = ax$, Which is not enough.
Any hints will be appericiated.