I have to find a sequence ${\{f\_k\}}\_{k = 1}^{\infty}$, with $f\_k : \mathbb{N} \to [0 , \infty)$, such that $\lim\_{k \to \infty} f\_k(n) = 0$ for all $n \in \mathbb{N}$ and $\int\_{\mathbb{N}} f\_k d \mu = 1$ for all $k \in \mathbb{N}$, where $\mu$ is the counting measure, i.e. $\sum\_{n = 1}^{\infty} f\_k(n) = 1$ for all $k \in \mathbb{N}$.

My main attempt was to write $f\_k(n) = \frac 1 {k 2^n}$, what gives me $\lim\_{k \to \infty} f\_k(n) = 0$ for all $n \in \mathbb{N}$, but $\sum\_{n = 1}^{\infty} f\_k(n) = \frac 1 k \neq 1$ if $k \neq 1$, then it is not valid. The two conditions seem to be incompatible.