# If $(a_n)$ is a decreasing sequence of strictly positive numbers and if $\sum{a_n}$ is convergent, show that $\lim{na_n}=0$ [duplicate]

If $(a_n)$ is a decreasing sequence of strictly positive numbers and if $\sum{a_n}$ is convergent, show that $\lim{na_n}=0$

Since $(a_n)$ is decreasing and bounded below, by Monotonic Convergence Theorem, $(a_n)$ converges. So, there exists $N$ such that $|a_n-L|< \epsilon$

Since $\sum{a_n}$ is convergent, by the Divergence test, we have $\lim_n{a_n}=0$, which means there exists $N$ such that $|a_n-0|< \epsilon$

Then I get stuck at here.

I try to figure out the statement's meaning by inserting some examples, like $a_n=\dfrac{1}{n^2}$

Can anyone guide me on this question?

• If you get stuck with Peter's suggestion, the full answer can be found here: math.stackexchange.com/a/354985/22064 Apr 22, 2013 at 20:05
• Isn't this a multiplicate?
– Did
Apr 22, 2013 at 20:07
• @Did Probably.
– Pedro
Apr 22, 2013 at 20:11
• Dec 26, 2013 at 14:46

Suppose that $$\lim\limits_{n\to\infty}na_n\ne0$$. That is, there is an $$\epsilon\gt0$$ so that for any $$N$$, there is an $$n\ge N$$ so that $$na_n\ge\epsilon$$. Since $$a_n$$ is decreasing, $$\sum_{k=n/2}^n a_k\ge\frac{n}{2}\frac{\epsilon}{n}=\frac{\epsilon}{2}$$ Since $$na_n$$ does not converge to $$0$$, we can find infinitely many such blocks from $$n/2$$ to $$n$$ that sum to at least $$\epsilon/2$$. Thus, we've shown the contrapositive: if $$\lim\limits_{n\to\infty}na_n\ne0$$, then $$\sum\limits_{k=0}^\infty a_n$$ diverges.

• How did you show $\lim_{n\to\infty}na_n$ exist? Feb 10, 2016 at 21:17
• I assumed that the limit was not $0$ and got a contradiction. Part of the limit not being $0$ is the limit not existing. Thus, the contradiction implies not only that the limit exists, but also that the limit is $0$.
– robjohn
Feb 11, 2016 at 0:24
• That answer is not satisfactory, we can only guarantee n_ja_{n_j}\geq \epsilon. Sep 15, 2017 at 23:21
• @checkmath: That is all one needs to deny convergence.
– robjohn
Sep 16, 2017 at 2:32

Hint Use Cauchy's criterion for the convergence of a series. Remember the sequence is decreasing and positive.

• Do you mean the Cauchy condensation test?
– user9464
Aug 13, 2013 at 18:31
• @Jack No. I mean Cauchy's criterion: a sequence of reals converges iff it is Cauchy.
– Pedro
Aug 13, 2013 at 18:57
• Ah, I see. So it's in @robjohn's answer, right?
– user9464
Aug 14, 2013 at 1:30
• @Jack I am suggesting you prove $p\to q$, not $\neg q\to\neg p$, but the idea is the same.
– Pedro
Aug 14, 2013 at 1:32

Given $\epsilon >0$, there exists an $N$ large enough such that $n,m\ge N \implies \sum a_k =|\sum a_k|<\epsilon$. Then for any $n\ge N$ we have that $$0\le na_{2n}\le a_n+a_{n-1}+ \ ... \ +a_{2n-1}=\sum a_k <\epsilon,$$ thus we have $\lim na_{2n}=0$.

Now fix $\epsilon >0$ and fix $N$ such that $n\ge N \implies na_{2n}=|na_{2n}|<\epsilon/3$. Now for $n>2N$ we can find an $m\ge n$ such that $n/3 \le m\le n/2$, hence we have $|ma_{2m}| <\epsilon/3$. Since $a_k$ is a decreasing sequence then we have that $|ma_n|\le |ma_{2m}|<\epsilon/3$, thus $|na_n|\le |3ma_n|<\epsilon$, since $n\le 3m$. Therefore $\lim na_n=0$.

Sorry if this is a bit messy. If you want a different proof or want me to add more detail, let me know.

• It should be $m\geq N$. Feb 11, 2014 at 17:04
• I would simplify the end, since you have that $2na_{2n}\to 0$ we only need to prove that $(2n+1)a_{2n+1}\to 0$ but that holds since $0\leq (2n+1)a_{2n+1}\leq (2n+1)a_{2n}=2na_{2n}+a_{2n}$ which go both to zero. Sep 15, 2017 at 23:49