how to disprove that a function is in the big O of another function

Question

how can we choose two functions f and g such that $\\$ f $\in O(g)$ But $\sum_0^n f(n) \notin O(n*g(n))$

Answer 1

Hint:   consider for example $\,f(n)=g(n)=\dfrac{1}{n+1}\,$.