# The series 1-3+1-3+1-3... is (c,1) summable?

To prove the series $$1-3+1-3+1-3+1-3+...$$ is (c,1) summable. A series $$\sum_{n=1}^\infty a_n$$ is said to be (c,1) summable if the sequence of partial sums $$s_n$$ is (c,1) summable.

A sequence $${s_n}$$ is said to be (c,1) summable to L if $$\lim_{n\to \infty} \sigma_n =L$$

Where $$\sigma_n = \frac{s_1+s_2+...+s_n}{n}$$

Now for our series the sequence of partial sums is $$1,-2,-1,-4,-3,-6,-5,...$$

Then sequence $$\sigma_n$$ would be $$1,-1,-2/3, -6/4, -9/5, -15/6,...$$

I couldn't conclude from this that the series is (c,1) summable i think these are unwanted calculations. I think we can't apply any rearrangement since the series diverges even still we can't conclude about the convergence of $$\sigma_n$$ . what am i missing here?

Is there any way that i can prove the (c,1) summability of the series?

• You have computed $\sigma_n$ incorrectly. For example, $\sigma_3 = \frac{1-3+1}{3} = -\frac13$, not $-\frac23$ as you have.
– MJD
Commented Aug 1, 2023 at 15:43
• @MJD actually $\sigma_n$ is the average of the sequence of partial sums. Not the average of the series!! Commented Aug 1, 2023 at 16:19

No, it is not (C,1) summable!

It is known that \begin{align} \liminf_n s_n\leq\liminf_n\sigma_n\leq\limsup_n\sigma_n\leq\limsup_ns_n\tag{0}\label{zero} \end{align} (see for example, Zygmund, A., Trigonometric Series, 3rd Edition, Volumes I & II, pp. 75 (Vol I), Cambridge University Press).

Notice that $$s_{2n}=-n$$ and $$s_{2n+1}=-n+1$$.

Comments: Inequality \eqref{zero} holds in the general setting of Toeplitz transformations in which $$\sigma_n=\sum^\infty_{m=1}a_{nm}s_m$$ where $$M=(a_{nm})$$, $$(n, m)\in\mathbb{N}\times\mathbb{N}$$, is in infinite matrix such that

• (0) $$a_{nm}\geq0$$ (this is the positive condition);
• (1) for any $$n\in\mathbb{N}$$, $$\sum^\infty_{m=1}a_{nm}<\infty$$;
• (2) for any $$m\in\mathbb{N}$$, $$\lim_{n\rightarrow\infty}a_{nm}=0$$;
• (3) $$\lim_n\sum^\infty_{m=1}a_{nm}=1$$.

The case $$a_{nm}=\frac1n\mathbb{1}(m\leq n)$$ corresponds to Cesaro summability $$(C,1)$$.

• i don't get access to the book u referred. Can you explain any further. In your explanation there is still a possibility that $\sigma_n$ may converge even if $s_n$ didn't. How do u conclude it is not (c,1) summable? Commented Aug 2, 2023 at 1:16
• @LakshmiPriya: for your particular series, $\lim_n\sigma_n=-\infty$. Commented Aug 2, 2023 at 2:39
• @LakshmiPriya: Here is a posting in MSE that touches on the inequality I stated in my posting. That inequality is more general and holds for positive Toeplitz transformations. Commented Aug 2, 2023 at 15:28