I have been asked to show that the series a_{n}=

\begin{matrix} \frac{n+1}{2}&if&n&is&odd& & \\ -\frac{n}{2}&if&n&is&even & & \end{matrix}

is not (C,1) Cesaro summable.

Not quite sure how to go about doing this so any help, hints, or process description will be appreciated.


Assuming the sequence starts at $n=1$, let us obtain an expression for the partial sums $$S_k = \sum_{n=1}^{k}a_n$$ If $k$ is even, then $$\begin{aligned} S_k &= \sum_{m=1}^{k/2}(a_{2m-1} + a_{2m}) \\ &= \sum_{m=1}^{k/2}\left( \frac{(2m-1)+1}{2} - \frac{2m}{2} \right) \\ &= 0 \end{aligned}$$ If $k$ is odd, then $$S_k = S_{k-1} + a_k = 0 + \frac{k+1}{2} = \frac{k+1}{2}$$ The first few $S_k$ are therefore $$1, 0, 2, 0, 3, 0, \ldots$$ The sum of the first $2n-1$ of these partial sums is simply $1 + 2 + \cdots + n = n(n+1)/2$. In other words, $$\sum_{k=1}^{2n-1}S_k = \frac{n(n+1)}{2}$$ and therefore the average of the first $2n-1$ partial sums is $$\frac{1}{2n-1}\sum_{k=1}^{2n-1}S_k = \frac{n(n+1)}{2(2n-1)}$$ which diverges to $\infty$ as $n \to \infty$.


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