Questions tagged [summation]

Question 1

Let $\displaystyle{n}\in\mathbb{N}^{+}$, show that: $${-\sum_{k=0}^{\color{red}{n}}(-1)^k\frac{(\pi/2)^{-2k+1}}{({\small2n-2k+1})!}\eta(2k)=}{\,\sum_{k=0}^{\color{red}{n-1}}(-1)^k\frac{(\pi/2)^{-2k}}{(...

Question 2

In the study of some lattice paths I came across the following identities. I would be interested in a simple algebraic proof. $\sum_{j=0}^{n}(2j)\binom{n-j}{k}\binom{n+j}{k}=(k+1)\binom{n}{k+1}\binom{...

Question 3

I was playing around with functions and got stuck trying to find the maximum value of a function. I started with $f(x)$ and $h(x)$, where $h(x)$ is the result after applying the sum of geometric ...

Question 4

Let $p$ be a prime, $1\le b \le p-1$ be an integer, and $\text{ord}(b)$ be the order of $b$ mod $p$. I am interested in the sum $$S_p(b) = \sum_{k=1}^{\text{ord}(b)}\frac{\sin\left(b^{k+1}\cdot\frac{p-...

Question 5

I am currently working my way through The Beginner’s Textbook for Fully Homomorphic Encryption by Ronny Ko. I cant wrap my head around how he grouped those terms up (p.100). If anybody could help me ...

Question 6

I am questioning a particular step of the solution presented to the following question: Cauchy’s root test for convergence states the following: Given a series $\sum_{k=1}^\infty a_k$, define $$\rho=\...

Question 7

This is a proof for the hermite identity: $$\sum_{k=0}^{n-1} \left\lfloor x + \frac{k}{n} \right\rfloor=\lfloor nx \rfloor$$ Combinatorial Interpretation of the RHS Let $S$ be the set of positive ...

Question 8

We have $\left\lfloor x\right\rfloor$ as the floor function and $\left\{x \right\}$ as the fractional part. Looking for the asymptotic expansion as $x \rightarrow \infty$ of $$DW (x)=\sum_{n \le x} \...

Question 9

Well, when I do the polynomial problem that my teacher gave me. I've tried new way to solve the problem by using p-adic $v_2$ but cannot solve it, here is the question: For two polynomials with ...

Question 10

I am doing a project and somehow the alternating sum $\sum_{i=0}^{n}(-1)^{i}{{n+i}\choose{i}}$ comes up. I am not not sure if there is any use of this sum, but just think it is an interesting sum ...

Question 11

This problem comes from the 1976 Putnam exam. Evaluate $$ L=\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n \left( \left\lfloor\frac{2n}{k}\right\rfloor -2\left\lfloor\frac{n}{k}\right\rfloor \right), $$ ...

Question 12

How can I prove that $\sum_{k=1}^{n} \left\lfloor \log_{2}\!\left(\frac{2n}{2k-1}\right) \right\rfloor = n$, where $n$ is a natural number? I discovered this identity while trying to prove Prove using ...

Question 13

I am trying to find a closed form for the following sum: $$ \sum_{k=0}^{n-1} \left( \frac{1}{(k+1)(n-k)} \cdot \binom{n+1}{k+1}^2 \right) $$ What I have tried so far I tried to simplify the expression ...

Question 14

Give a combinatorial proof that \begin{eqnarray*} \sum_{i+j+k=n} \binom{2i}{i} \binom{2j}{j} \binom{2k}{k} = (2n+1) \binom{2n}{n}. \end{eqnarray*} Where did this come from ? ... In this question (...

Question 15

I’m trying to understand how to recognize when a series is telescoping. Consider the series $$ \sum_{n=3}^{\infty} \frac{1}{n(n-1)(n-2)}. $$ Using partial fraction decomposition, we get $$ \frac{1}{n(...

Stack Exchange Network

Questions tagged [summation]

On $\sum_{k=0}^{n-1}\,(-1)^k\,\frac{(\pi/2)^{-2k}}{(2n-2k)!}\,\beta(2k+1)$

Proof of some binomial identities [closed]

How to find the maximum value of $g(x)=|f(x)−h(x)|$ where $f(x)$ is a finite geometric series? [closed]

Primes for which sum of sine ratios is nearly always an integer

Grouping of Polynomial Ring Calculation [duplicate]

Understanding the proof of Cauchy's root test.

Question regarding a combinatorial map between two sets for proving hermite identity

Asymptotic expansion as $x \rightarrow \infty$ of $\sum_{n \le x} \lfloor\frac{x}{n}\rfloor \left\{{2\sqrt{\lfloor\frac{x}{n}\rfloor}}\right\}$

How to prove this inequality $2^n\le 2t+v_2\left( \sum_{i=t}^{2^{n-1}} \binom{2^n}{2i}\binom{i}{t}\right)$ $(1\le t\le 2^{n-1}$)?

Can we simplify the alternating sum $\sum_{i=0}^{n}(-1)^{i}{{n+i}\choose{i}}$?

Limit with floor sums reminiscent of the exponent of the central binomial coefficient

Prove that $\sum_{k=1}^{n} \left\lfloor \log_{2}\!\left(\frac{2n}{2k-1}\right) \right\rfloor = n$ where $n$ is a natural number.

Closed form for a symmetric sum of squared binomials

Proof of a formula involving central binomial coefficients.

Telescopic series: how to identify it after breaking it down into partial fractions?

Hot Network Questions