Proof of convergence of series (progression) First, I'm a beginner in this site. In addition, my mother tongue is not English. Thus, I'm sorry if the sentences I write are difficult to understand.
I'll move on to the main topic.
I can't solve this problem.
Precondition(Given);
There exists a progression ${x_n}$ such that all terms in the progression are $0$ or more($x_n \geq 0$).
Moreover, an infinite series $\sum_{n=1}^∞ n^2・(x_n)^2$ converges.
Problem;
If the above precondition works, prove the fact that $\sum_{n=1}^∞ x_n$ converges.
I think that I have to determine the condition for the fact that an infinite series $\sum_{n=1}^∞ n^2・(x_n)^2$ converges, and have to use the condition in order to prove the fact that $\sum_{n=1}^∞ x_n$ converges.
However, I can't solve this problem.
If you find how to solve this problem, I want you to teach it.
 A: The series $\sum_{n=1}^{\infty}1/n^2$ converges to $K\in \Bbb R+$. (We do not need to know what $K$ is.)
Let $x_n=y_n/n.$ By the Cauchy-Schwartz Inequality, if $M\in \Bbb Z^+$ then $$\sum_{n=1}^Mx_n=\sum_{n=1}^M(1/n)y_n\le$$ $$\le \left(\sum_{n=1}^M1/n^2\right)^{1/2}\cdot \left(\sum_{n=1}^My_n^2\right)^{1/2}\le$$ $$\le \sqrt K \cdot \left(\sum_{n=1}^My_n^2\right)^{1/2}=$$ $$=\sqrt K\cdot 
\left(\sum_{n=1}^Mn^2x_n^2\right)^{1/2}.$$
The Cauchy-Schwartz Inequality: $(\,\sum_{n=1}^Mw_n^2 \,)\cdot (\,\sum_{n=1}^My_n^2\,)-(\sum_{n=1}^Mw_ny_n)^2=$ $=\sum_{1\le i\le j\le M}(w_iy_j-w_jy_i)^2 \ge 0.$
A: Consider the p-series with $p=\frac{3}{2}$. Well, because $\sum_{0}^{\infty}n^2\left(\frac{1}{n^{3/2}}\right)^2$ diverges then we know that $x_n$ as $n$ goes to infinity must be smaller than the value of the p-series with $p=\frac{3}{2}$. Well, we know that this p-series converges. Thus by the comparison test, because the p-series converges so does the summation $\sum_{0}^{\infty}x_n$
A: Hint: We want show that $\sum x_n$ converges and we're given the convergence of a series with squared terms. This strongly suggests using the Cauchy-Schwarz Inequality.
How can you use the Cauchy-Schwarz Inequality to relate the series $\sum x_n$ to the series $\sum n^2 x_n^2$?
