# Questions tagged [cesaro-summable]

For questions about Cesàro summation and Cesàro summable sequences.

148 questions
Filter by
Sorted by
Tagged with
1 vote
73 views

• 1,096
34 views

### Convergence speed of Cesaro mean

Consider a sequence $(a_n)$ satisfying $\lim_{n\to\infty} a_n = a$. Let $b_n = \frac{1}{n} \sum_{i=1}^n a_i$. I have already known that $\lim_{n\to\infty} b_n = a$. I am wondering is there any ...
• 43
32 views

### Is the average of two convergent series equal to the Cesàro sum of the alternating series?

If we have $(a_n)$ and $(b_n)$ such that $\sum a_n$ converges and $\sum b_n$ converges, I know that we do not necessarily have that $\sum c_n$ (where $(c_n)_n=a_0,b_0,a_1,b_1,\dots$ converges. But is ...
47 views

### Convergence of Power Series to Its Cesaro Sum sentence in Fourier proof

Well, I am learning about Fourier sum, and I encountered Cesaro sum in the proof of convergent uniformly of Fourier sum, I know that Fejér sentence says that: $\|f(x) - \sigma_n(f))\| < \epsilon .$ ...
47 views

• 135
159 views

• 3,751
87 views

### Weakly null sequence in Schreier space

Define the Schreier space $X:=\overline{c_{00}}^{\Vert\cdot\Vert}$ where $$\Vert x\Vert = \sup\bigg\{\sum_{i=1}^k|x_{n_i}|:k\le n_1<n_2\cdots <n_k\bigg\}.$$ Show that there exists a weakly null ...
• 453
42 views

### How to prove the Divergence of cesaro mean of harmonic series [duplicate]

How do you prove that the Cesàro mean of the harmonic series diverges. I know that the harmonic serie $$\sum_{k=1}^{\infty} \frac{1}{k}$$ diverges, However, I'm having trouble proving that it also ...
• 593
30 views

• 1,016
71 views

### Equivalence on Cesàro convergence

Let $(x_n)$ be a bounded real sequence. Then $$\lim_{n\to \infty}\frac{1}{n}\sum_{i=1}^n x_i=0 \Longleftrightarrow \lim_{n\to \infty}\frac{1}{2^n}\sum_{i=2^n+1}^{2^{n+1}} x_i=0.$$ Do you have a ...
• 15.4k
1 vote
78 views

• 321
187 views

Let $(e_n)_{n\in\mathbb{\mathbb{Z}}} = (t\mapsto e^{int})_{n \in \mathbb{Z}}$ the Fourier orthonormal base of $L^2(\mathbb{T})$ with the scalar product $\langle f,g\rangle = \int_{-\pi}^\pi f(t)\bar{g}... • 5,695 0 votes 0 answers 53 views ### Generalized Cesàro summability of$(-1)^nn^p$A sequence$\{a_n\}_{n\geq 0}$is said to be$(C,\alpha)$-summable if$\lim_{n\to\infty} S^\alpha_n$exists, where $$S^n_\alpha = \sum_{k=0}^n \frac{ {n \choose k} }{ {n+\alpha \choose k} } a_k.$$ ... • 389 4 votes 3 answers 187 views ### Proof that$\sum_{n=0}^{\infty}(-1)^{n} = \frac{1}{2}$. Is there any error? So, I proved that: $$\int f(\ln x)\ dx = x \sum_{n=0}^{\infty}(-1)^{n} f^{(n)}(\ln x) \ \ \ +\ \ C$$ where$f^{(n)}$is the nth derivative of$f$. if we let$f(x) = e^{x}$then$f^{(n)}(x) = e^x$as ... • 4,868 1 vote 0 answers 205 views ### Cesàro Mean of Convergent Subsequence Converges? Let$X$be a compact and convex subset of$\mathbb R^n$and let${(x_{t})}_{t}$be a sequence in$X$such that$||x_{t+1}-x_{t}||_{\infty}\leq \frac{1}{t+1}$for all$t=0,1,...$. Let${(x_{t_n})}_{...
Loosely, I would like to know "how many" binary sequences are non-convergent in the sense of their average. Let $\{0,1\}^{\mathbb N}$ denote the set of all binary sequences. This set is homeomorphic ...