I apologize if this is in the wrong subject, but it is for my calculus class.

(a) Let $\displaystyle a_n >= 0, n \in N$ Let $\displaystyle s_n = a_1 + a_2 + · · · a_n$, i.e., partial sums of $\displaystyle \sum a_n$ from n = 1 to infinity.

Furthermore, let

$\displaystyle \tau_n = \frac{s_1 + s_2 + ... + s_n}{n}$ $\displaystyle , n \in N.$

Prove:

If $\displaystyle lim s_n = s $ then $\displaystyle lim \tau_n = s $ as well.

(b) The $\displaystyle \tau_n$ are called Cesaro sums, and the process is called Cesaro summation. It is used when the original sequence converges slowly or when a series does not have convergent partial sums. For example the series:

$\displaystyle 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, . . .$

$\displaystyle s_n $ does not have convergent partial sums but $\displaystyle \tau$ does converge. Find the Cesaro limit ( $\displaystyle lim \tau_n$)

of the sequence.