# Sequence...

#### Also sprach Zarathustra

Let $$\displaystyle (a_n)^\infty_{n=1}$$ converges to $$\displaystyle a$$.

Is the next sequence $$\displaystyle b_n$$ also converges?

$$\displaystyle b_n=\frac{a_1^2+a_2^2+\dots +a_n^2}{n}$$

Thanks!

#### Plato

MHF Helper
Let $$\displaystyle (a_n)^\infty_{n=1}$$ converges to $$\displaystyle a$$.
Is the next sequence $$\displaystyle b_n$$ also converges?
$$\displaystyle b_n=\frac{a_1^2+a_2^2+\dots +a_n^2}{n}$$
Do you know the idea of the sequence of means?
Because $$\displaystyle \left(a_n\right)$$ converges it is true that $$\displaystyle \left(a_n^2\right)$$ also converges.