39 questions linked to/from Series converges implies $\lim{n a_n} = 0$
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### 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 ...
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### Prove $\lim\limits_{n\to\infty}na_n\ln(n)=0$ [duplicate]

Let $a_n>0$,$\sum a_n$ is convergent,$na_n$ is monotone, Prove: $$\lim\limits_{n\to\infty}na_n\ln(n)=0$$ I try to prove $a_n=o(n\ln(n))$, but it doesn't work.
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Let $(u_n)$ be a non increasing sequence of reals such that $\sum u_n$ converges. I'm investigating the question: do we always have $u_n = o(1/n)$? I suspect not and I'm trying to build a sequence $(... 0answers 39 views ### If$a_n \searrow 0$and$\sum a_n$converges, is it true that$n a_n \to 0$? [duplicate] I am posting for a friend: he has a strong feeling that the following should be true but can't prove it: "Let$(a_n)$be a real sequence. If$a_n \searrow 0$and$\sum a_n$converges, then$n a_n \to ...
I'm trying to show this: If $\Sigma_{n=1}^{\infty} {a_n}$ is a convergent series, and $a_n \geq a_{n+1}$ then $\lim\limits_{n \to \infty} {na_n} =0$. I already prove that the sequence $\{ na_n \}$ ...
Let $\{a_n\}$ be a decreasing positive sequence. Suppose that $\sum a_n$ is convergent. Q. Is $na_n$ convergent to $0$.