Problem 203 (difficulty: 4/10)

Consider the sequence \(\displaystyle s_n\) of arithmetic means

\(\displaystyle s_n=\frac{a_1+\ldots +a_n}{n} \)

corresponding to the sequence \(\displaystyle a_n\). Show that if \(\displaystyle \lim\limits_{n\to \infty} a_n=a,\) then \(\displaystyle \lim\limits_{n\to\infty} s_n =a.\) Give an example when \(\displaystyle (s_n)\) is convergent, but \(\displaystyle (a_n)\) is divergent.

Solution:

\(\displaystyle n>n_0(\varepsilon/2)\) we have \(\displaystyle |s_n-a|=\left|\frac{(a_1-a)+\cdots+(a_{n_0}-a)}n + \frac{(a_{n_0+1}-a)+\cdots+(a_{n}-a)}n\right|\).

Let \(\displaystyle n_1>\frac{(a_1-a)+\cdots+(a_{n_0}-a)}{\varepsilon/2}\). Then for \(\displaystyle n>n_0,n_1\) \(\displaystyle |s_n-a|<\varepsilon/2+\varepsilon/2\).

Give me another random problem!