My question may not make much sense because I'm still trying to wrap my mind around infinite sequences and series. I seem to have good working knowledge of when and why to apply a certain tests for a given series, but something just seems like it is missing in my understanding.
Conceptually, why can a sequence converge at any number (except $-\infty$ and $\infty$, which aren't numbers anyway), but the limit of a series must approach $0$ to converge?
Is the reason behind the requirement for the limit of a series approaching $0$ to converge because the series eventually stops summing numbered terms infinitely due to its convergence to a specific number? (Sorry if this question makes no sense)!
And is it acceptable for a sequence to approach any number simply because the sequence isn't being summed?