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Let $(x_n)$ a decreasing sequence and $\sum x_n\to s$. Then $(n\cdot x_n)\to 0$

Check my proof please, Im not completely sure about it correctness.

If $\sum_{k=h}^\infty x_k= s$ then we can rewrite the sum for starting index $1$ with the change $k-h=j$, then

$$\left(\sum_{j=1}^n x_j\right)-n\cdot x_n=\sum_{j=1}^n (x_j-x_n)$$

Then taking limits

$$\color{red}{\lim_{n\to\infty}\left[\left(\sum_{j=1}^n x_j\right)-n\cdot x_n\right]}=\lim_{n\to\infty}\sum_{j=1}^n (x_j-x_n)=\sum_{j=1}^\infty (x_j-0)=\color{red}{\lim_{n\to\infty}\sum_{j=1}^n x_j}=s$$

where I used the fact that $(x_n)\to 0$. Thus equating the colored expressions this implies that $\lim_{n\to\infty} nx_n=0$.

The proof, to my eyes, seems correct but I dont needed in any moment to use the fact that $(x_n)$ is a monotonic sequence so it is possible that I make a mistake somewhere or that the proof is incorrect.

My second attempt

Because $\sum x_k$ converges and is positive (cause $(x_n)\downarrow 0$) we can write

$$\sum_{k=n+1}^{2n+m}x_k=\left|\sum_{k=n+1}^{2n+m}x_k\right|<\epsilon/2,\quad \forall n,m\ge N$$

Then, cause $(x_n)$ is decreasing

$$(n+m)x_{2n+m}\le\sum_{k=n+1}^{2n+m}x_k<\epsilon/2\\\implies (2n+m)x_{2n+m}\le2(n+m)x_{2n+m}<\epsilon,\quad\forall n,m\ge N$$

Because $m$ is arbitrary setting $M=2N>N$ we can finally write

$$nx_n<\epsilon,\quad\forall n\ge M$$

It is this proof correct? Thank you.

Masacroso
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  • The step $$ \lim_{n \to \infty}\sum_{j=1}^n (x_j - x_n) = \lim_{n \to \infty}\sum_{j=1}^n \lim_{n \to \infty}(x_j - x_n) $$ Seems a bit suspect, but I can't say with certainty that it's wrong – Ben Grossmann Oct 25 '16 at 00:07
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    If $x_n=\frac{(-1)^n}n$, $\sum_nx_n$ converges, and $x_n\to 0$ (though not monotonically), but $nx_n=(-1)^n$ does not converge to $0$. This shows that your argument can’t be right, since nothing in it rules out this example. – Brian M. Scott Oct 25 '16 at 00:17
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    @Omnomnomnom. It IS a wrong step. The attempted proof does not use the monotonicity of $x_n$. But the result does not hold for all non-monotonic $x_n$. E.g. if $x_{(n^4)}=1/n^2$, and $x_n=0$ when $n$ is not a $4$th power. – DanielWainfleet Oct 25 '16 at 00:22
  • Ok, it is clear that this proof is not correct. I will investigate a different proof, this time totally formal. Thank you to all. – Masacroso Oct 25 '16 at 00:23
  • On top of my head, I think your proof can be adjusted with Beppo-Levi theorem. However, it is not a result which is taught in the first course of analysis. –  Oct 25 '16 at 00:25
  • It's just a limit comparison test with $\sum_j 1/j$ – Ben Grossmann Oct 25 '16 at 01:43

2 Answers2

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Hint:

If $\sum_{k=1}^\infty x_k = s,$ then by the Cauchy criterion for any $\epsilon > 0$ there exists $N \in \mathbb{N}$ such that if $n > N$ we have $\sum_{k= n+1}^{2n} x_k < \epsilon.$ Since $(x_n)$ is decreasing and positive,$\sum_{k= n+1}^{2n} x_k > nx_{2n}.$

RRL
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  • Please say to me if my above proof, based on that, is correct. – Masacroso Oct 25 '16 at 11:53
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    @Masacroso: You might add -- with $m=0$ we see $2nx_{2n} \to 0$ and with $m = 1$ we see $(2n+1)x_{2n+1} \to 0$. Since both subsequences converge it follows that $nx_n \to 0$. Well done. – RRL Oct 25 '16 at 14:22
  • Doesn't it follow from this that $f(n)a_n \to 0$ for any $f(n) \geq 1$? – Georgi Kocharyan May 11 '21 at 09:56
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Here's a correct proof: apply summation by parts to $\sum a_n b_n$, where $b_n = 1$ for all $n$.

Ben Grossmann
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