Convergence in $L^2$ doesn't imply uniform convergence

Question

I am working on implications between different types of convergences in series of functions. I think that $L^2$ convergence doesn't imply uniform convergence, however I cannot find a counter-example. I cannot picture how a series of uniform convergent functions looks like, and this is my main issue here. Can someone give me a hint? (Both on the counter-example and on the idea of how these series look like)