Why can the limit of a sequence approach a number and converge, but the limit of the series must approach $0$ to converge?

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?

• You mean thr limit of the summand – Mhenni Benghorbal Dec 4 '13 at 15:17
• Try to look at the partial sums of the series when $a_n$ does not converge to $0$ for $n\rightarrow \infty$. – user112167 Dec 4 '13 at 15:17
• @MhenniBenghorbal excuse my ignorance, but what is a summand? – hax0r_n_code Dec 4 '13 at 15:18

I think you are mixing sequences and their sums. The terminology is not very helpful here, I would suggest the following.

Let $a_k$ be any sequence. Then you can define the sequence of partial sums: $$S_n = \sum_{k=1}^n a_k.$$ Then $$\sum_{k=1}^\infty a_k = \lim_{n\to \infty} S_n.$$

Given any sequence $S_n$ you can find some $a_k$ such that $S_n$ are the partial sums of $a_k$.

So you see that a series may have any limit, as the sequence do. However if a series $S_n$ has a finite limit, then the corresponding sequence $a_k$ must tend to zero.

Don't confuse the series $S_n$ with its general term $a_k$.

• So is it accurate to say that a series is simply a sequence of partial sums from a given sequence? – hax0r_n_code Dec 4 '13 at 15:29
• That is my preferred point of view. – Emanuele Paolini Dec 4 '13 at 15:31
• That's interesting because I never could see the relationship between a sequence and a series. – hax0r_n_code Dec 4 '13 at 15:37
• I say that the terminology does not help because formally the expression: $\sum_{k=1}^\infty a_k$ is a number. So it makes no sense to say that such expression converges or diverges. But really you are referring to the sequence of partial sums, not to the sum itself. – Emanuele Paolini Dec 4 '13 at 15:40
• @EmanuelePaolini It makes perfect sense to say that expression converges or diverges. It is nothing but a limit, and limits can sometimes fail to be numbers, but instead diverge. – Potato Dec 4 '13 at 16:05