Consider the sequence of real numbers $\{x_n\}_{n=1}^{\infty}$ such that:
$$\lim_{n \rightarrow +\infty} \frac{x_1^{2p}+x_2^{2p}+\cdots+x_n^{2p}}{n}=0$$.
where $p$ is a positive integer. Prove that $\lim \limits_{n \rightarrow +\infty} \frac{x_1+x_2+\cdots+x_n}{n}=0$.
Does the converse hold?
Solution
We recall the inequality:
$$\left ( \frac{1}{n}\sum_{k=1}^{n}x_k \right )^{2p}\leq \frac{1}{n}\sum_{k=1}^{n}x_k^{2p} \Rightarrow \left | \frac{1}{n}\sum_{k=1}^{n}x_k \right |\leq \sqrt[2p]{\frac{1}{n}\sum_{k=1}^{n}x_k^{2p}} \longrightarrow 0$$
proving the first part.
The converse does not hold. Indeed take $x_n=(-1)^n$ thus:
$$\frac{1}{n}\sum_{k=1}^{n}x_k = \left\{\begin{matrix}
0 &, &\text{if n is even} \\
-\frac{1}{n}&, &\text{if n is odd}
\end{matrix}\right.$$
Hence $\lim \limits_{n \rightarrow +\infty} \frac{x_1+x_2+\cdots+x_n}{n}=0$ but $\lim \limits_{n \rightarrow +\infty} \frac{x_1^{2p}+x_2^{2p}+\cdots+x_n^{2p}}{n}=1$.
$$\lim_{n \rightarrow +\infty} \frac{x_1^{2p}+x_2^{2p}+\cdots+x_n^{2p}}{n}=0$$.
where $p$ is a positive integer. Prove that $\lim \limits_{n \rightarrow +\infty} \frac{x_1+x_2+\cdots+x_n}{n}=0$.
Does the converse hold?
Solution
We recall the inequality:
$$\left ( \frac{1}{n}\sum_{k=1}^{n}x_k \right )^{2p}\leq \frac{1}{n}\sum_{k=1}^{n}x_k^{2p} \Rightarrow \left | \frac{1}{n}\sum_{k=1}^{n}x_k \right |\leq \sqrt[2p]{\frac{1}{n}\sum_{k=1}^{n}x_k^{2p}} \longrightarrow 0$$
proving the first part.
The converse does not hold. Indeed take $x_n=(-1)^n$ thus:
$$\frac{1}{n}\sum_{k=1}^{n}x_k = \left\{\begin{matrix}
0 &, &\text{if n is even} \\
-\frac{1}{n}&, &\text{if n is odd}
\end{matrix}\right.$$
Hence $\lim \limits_{n \rightarrow +\infty} \frac{x_1+x_2+\cdots+x_n}{n}=0$ but $\lim \limits_{n \rightarrow +\infty} \frac{x_1^{2p}+x_2^{2p}+\cdots+x_n^{2p}}{n}=1$.
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