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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?

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

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    $\begingroup$ So is it accurate to say that a series is simply a sequence of partial sums from a given sequence? $\endgroup$ – hax0r_n_code Dec 4 '13 at 15:29
  • $\begingroup$ That is my preferred point of view. $\endgroup$ – Emanuele Paolini Dec 4 '13 at 15:31
  • $\begingroup$ That's interesting because I never could see the relationship between a sequence and a series. $\endgroup$ – hax0r_n_code Dec 4 '13 at 15:37
  • $\begingroup$ 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. $\endgroup$ – Emanuele Paolini Dec 4 '13 at 15:40
  • $\begingroup$ @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. $\endgroup$ – Potato Dec 4 '13 at 16:05

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