Questions tagged [summation-by-parts]

Question 1

The following inequality I found on an inequalities cheat sheet should follow from summation by parts, according to my search: if $b_1\ge b_2\ge\ldots\ge b_n\ge 0$, then $\sum_{k=1}^na_kb_k\ge b_1\...

Question 2

I'm trying to use the specific form of Abel's sum formula: $\sum_{n \leqslant x} a_n f(n) = A(x)f(x) - \int_1^x A(t)f'(t)\,dt$ where $A(x) = \sum_{n \leqslant x} a_n$ for the following example: $A(x)=\...

Question 3

Edit: my post was marked as a duplicate of this post. The answers to this post are not related to what I'm asking. I am not looking to use the Cauchy-Schwarz inequality, as I mentioned in my post. I'm ...

Question 4

$\textbf{Question}$： Let $\{z_n\}_{n=1}^{\infty}$ be any complex series while $\sum\limits_{n=1}^{\infty}\left|z_{n+1}^{-1}-z_n^{-1}\right|=\infty$, suppose $\sum\limits_{n=1}^{\infty}a_n z_n$ is ...

Question 5

Let $\chi$ be a Dirichlet character modulo $q$, and consider the Dirichlet series $$L(s,\chi) = \sum_n \frac{\chi(n)}{n^s}$$ We know that $\sum_{k < n} \chi(k) = o(n)$ for $n > q$ for instance, ...

Question 6

Lets say we are having 4 "N" bit values, and in each of them there is exactly 1 10-bit pattern of "1000000001" somewhere (x can be 0 or 1, and it will be exactly N-10) - ...

Question 7

Let $(a_n)_{n \in \mathbb{N}}$ be a sequence of complex numbers. Furthermore, suppose that exists some $z_0 \in \mathbb{C}$ such that $\sum_{n=1}^{\infty}\frac{a_n}{n^{z_0}}$ converges. Now, my goal ...

Question 8

I'd like to find a closed-form formula for $$\sum_{k=1}^{n-2} [(-k + (-1)^k)^2 - 3]$$ using the indefinite summation method. That would be fine besides this annoying factor $(-1)^k$ for the expanded ...

Question 9

Let f be an arithmetic function and let $F(s)$ be its Dirichlet series $\sum_{n=1}^{\infty}f(n)n^{-s}$. We say f has an analytic mean value A if $F(s)=\frac{A}{s-1}+o(\frac{1}{s-1})$ as $s\rightarrow ...

Question 10

It is easy to prove that for $0<q<1$, $\sum_{k=0}^{n} q^{(2k+1)^{3/2}}$ converges because it is always increasing and it is always smaller than the convergent series $\sum_{k=0}^{n} q^{2k+1}$ ...

Question 11

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of indepent random variables all uniformly distributed on $[-1,1]$ Calculate the probability that $$\sum_{n=1}^{\infty}{\frac{X_n}{n}}$$ converges Using ...

Question 12

This is something I stumbled upon a few years back, and has a geometrically intuitive explanation, as well as being useful in summing powers of integers. However, with some occasional attempts at an ...

Question 13

Given function of $\tau(n,\vartheta)$ denotes $\sum_{d|n}d^{i\vartheta}$ for $\vartheta\neq 0$. Also, here $e(x)=\exp(2\pi ix)$. The book mentioned the theorem to be used for the exercise (problem ...

Question 14

Given a sequence $\left(a_n \right)$, the operators $\Delta, E$ are defined by $$\Delta a_n=a_{n+1}-a_n,\\Ea_n=a_{n+1}.$$ Let $\left(u_n \right)$ and $\left(v_n \right)$ be two sequences. Let $\...

Question 15

Suppose that $\lambda_1(A) \geq \lambda_2(A) \geq \cdots \geq \lambda_n(A)$ are (real) eigenvalues of a Hermitian matrix $A$ and denote the empirical measure by $L_A:= \frac{1}{n}\sum_{i=1}^n \delta_{\...

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Questions tagged [summation-by-parts]

On an application of summation by parts

Help with Abel's sum formula example.

If $\sum a_n$ converges and each $a_n \geq 0$, $\sum \frac{\sqrt{a_n}}{n}$ converges using summation by parts

Limitation about the quotient of two complex series

Summation by parts and small sums

Binary combination - find individual parts from sum

Dirichlet's series

Antidifference of alternating sequence for $\sum_{k=1}^{n-2} [(-k + (-1)^k)^2 - 3]$

Prove that the analytic mean value of an arithmetic function equals the logarithmic mean value

limit of $\sum_{k=0}^{n} q^{(2k+1)^{3/2}}$ with $0<q<1$?

Probability that $\sum_{n=1}^{\infty}{\frac{X_n}{n}}$ converges where $(X_n)_{n \in \mathbb{N}}$ is a sequence of i.i.d with $X_n ~ Unif(-1,1)$

Proof that $\sum_{i=0}^n i^k = \sum_{i=0}^n (n-i)((i+1)^k-i^k)$

The bound on the summatory function of $\tau(n,\vartheta)$ using Van de Corput's method

Order of Operators - Question in Proof of Abel's Transformation - Summation by Parts

Integration by parts involving empirical/counting measure

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