Summarise with Cesaro summation

Question

Lets consider series $A = \sum_{n=0}^\infty(-1)^n$. $A = 1 - 1 + 1 - 1 \dots$

Lets consider k-th term of series A. Move it to $2^kth$ position. So, and repeat it for every term in series. Obviously, there are positions in new series which are empty. Denote their by zero.

So, what is the sum of new series if we summarise with Cesaro summation?

Answer 1

Hint: the partial sums $S_n$ can be $1$ or $0$. What will be $S_n/n$?