Is 1/n gradually approaching 0 when n approaches infinity.

Question

According to the definition of Convergent series, "the series is said to be Convergent when n approaches infinite an=L, where L is a constant." And when we prove that n/(n+1) will approach 1, we convert it to 1/(1+1/n) because 1/n will gradually approach 0. But in the P series, when P=1, it is a divergent (I know its proof process). Isn't the nth item here close to 0? Does this mean that I have any misunderstandings about the definition of convergent series. Or is the definition of harmonic series not applicable to general conversion series?

Thanks

Answer 1

Series and sequence are synonyms in everyday English, but in math they have different meanings.

The sequence whose $n$th term is $\frac{1}{n}$ does in fact converge to zero. But the series whose $n$th term is $\frac{1}{n}$ is the same as the sequence whose $n$th term is $\sum_{k=1}^n \frac{1}{k}$. That sequence does not converge, so the series $\sum_{n=1}^\infty \frac{1}{n}$ does not converge.

Because the phrases and notations around limits and sums are cumbersome, they often (unfortunately) get omitted in conversation. This can lead to some contradictory abbreviations:

• Because $\displaystyle\lim_{n\to \infty} \frac{1}{n} = 0$, one might say “$\frac{1}{n}$ converges”. This is sloppy, but I think conventionally understood to mean the sequence converges.
• Because $\displaystyle\lim_{N\to \infty} \sum_{n=1}^N \frac{1}{n} = \infty$, one might be tempted to say “$\frac{1}{n}$ diverges.” You see the ambivalence, which is what brought you to this question.

Try to avoid these abbreviations, or each time you see them, get some clarity: “Do you mean the sequence $\frac{1}{n}$ or the series of the terms in that sequence?”

Answer 2

Try to avoid the phrasing "approaching" because there is nothing that moves. A limit is a number such that every neighborhood of it no matter how small it is contains all but finitely many members of the sequence. The distinction and the avoidance of the term approaching are important because a limit is a fixed certain number and nothing anything approaches.

A series is considered converging if its sequence of partial sums $$s_n:=\sum_{k=1}^n a_k$$ is convergent. So the term convergence is reduced to those of sequences, even in the case of series.