
Cesaro Sum Help
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.

I figured out how to do part b (compute the Cesaro sum), so now I only need help on
Prove:
If http://www.mathhelpforum.com/mathhe...f37407e81.gif then http://www.mathhelpforum.com/mathhe...e601e3f41.gif as well.