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Questions tagged [summation]

Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.

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Good upper bound for $\sum_{n=1}^N a^{-n} n^{-b}$, for $a, b \in (0, 1]$

Let $a, b \in (0, 1]$ and define $S_N:=\sum_{n=1}^N a^{-n} n^{-b}$. Question What is a good upper bound for $S_N$ ? Observations By a simple (and probably careless) application of Cauchy-Schwarz, ...
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1answer
13 views

Sum of product of Stirling numbers

We have for $n>0$, $k>0$ $$\sum\limits_{j=1}^{\min(n,k)}(j!)^2{n\brace j}{k+1\brace j+1}=\sum\limits_{j=0}^{\min(n,k-1)}j!(j+1)!{n+1\brace j+1}{k\brace j+1}$$ How can we prove it?
4
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1answer
27 views

Combinatorial proof for $\sum_{k=0}^p (-1)^k {n \choose k} = (-1)^p {n-1 \choose p}$

I am trying to give a combinatorial proof for: $$\sum_{k=0}^p (-1)^k {n \choose k} = (-1)^p {n-1 \choose p}$$ Where $p$ and $n$ are natural numbers. We could easily see that if $p=n$ this reduces ...
2
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3answers
29 views

$\sum_{m=0}^N \frac{1}{m+1}{N\choose m}p^m (1-p)^{N-m}=\frac{1-(1-p)^{N+1}}{(N+1)p}$

While reading a journal article, I noticed an equation which looked like the sum of the average binomial coefficient. But, I have no idea how equation was derived. $$\sum_{m=0}^N \frac{1}{m+1}{N\...
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0answers
17 views

Solution for a constrained (v 1st & u 2nd) sum $\sum_{a=1, |a*u+v|\le N}^{N}\left(2\Bigl\lfloor{N/a}\Bigr\rfloor+1\right)\sum_{a=1,|a*u+v|\le N}^{N}N$

I am summing over the multiplication of two polynomials (a linear times a quadratic) that results in the following constrained (over coefficients) double sum $$\sum_{a=1}^{N}\left(2\Bigl\lfloor{\frac{...
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1answer
24 views

Hurwitz zeta function for $s=0$ $\zeta(0,1/2)$

I'm studying the Casimir Effect with perfect spherical boundary which involves the use of the Hurwitz zeta function. I've been staring for a while at this equation: \begin{align} \sum_{l=1}^{\...
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0answers
106 views

Triple sum $\sum\limits_{a=1}^{\infty} \sum\limits_{b=1}^{\infty} \sum\limits_{c=1}^{\infty} \frac{\cos a \cos b \cos c}{a^2 + b^2 + c^2}$

We have poor water heating system in our countryside house (currently it takes 4 hours to warm up the water), and my father has decided to improve it; he bought a water tank and placed it up in the ...
2
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1answer
31 views

Proving the summation of a double factorial infinite series.

$$ \sum_{n=0}^{\infty }\frac{(-1)^{n}((2n-1)!!)^2}{(2n)! (2^{2n})} = \frac{2}{\sqrt{5}} $$ I came across this summation through some other work, came across the solution as part of a function, but I ...
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1answer
23 views

Find the summation of given expression

I am trying to solve the following question which is Ex3 from Arthur Engel, Problem Solving strategies. Here is the question: $\sum_{k=1}^n k^3 {n \choose k}$ and asks to find the sum. I am sincerely ...
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1answer
30 views

Is it possible to find a closed form $\sum_{i=0}^n x^{f(i)}$ in general? For $f : i \mapsto i + i^2$?

Let be $f : \mathbb{N} \to \mathbb{N}$, I'm interested if it is possible to find a closed form of $\displaystyle \sum_{n=0}^{p} x^{f(n)}$ for all $x \in \mathbb{C}$ for all $p \in \mathbb{N}$, also ...
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1answer
48 views

Give a closed formula for the recursive series of $S_1 = \frac{a_1}{b_1}, S_{n+1} = \frac{a_{n+1}}{b_{n+1}+S_n}$

The Problem: The following real numbers are given: $$a_1,a_2,a_3,...\in\Bbb{R}\backslash\{0\} \\ b_1,b_2,b_3,...\in\Bbb{R}\backslash\{0\}$$ We define a recursive series of: $$S_1 = \frac{a_1}{b_1} ...
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0answers
53 views

When is $a(n)$ prime?

Question: When is $a(n)\in P$ compared to all possible values of $n$? where $P$ denotes the set of primes. What is the density of the primes in the sequence? Consider the sum of the prime counting ...
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3answers
48 views

Prove that $\sum_{n=0}^\infty {\frac{x^{2n+1}}{(2n+1)!}} = \frac{e^x - e^{-x}}{2}$

I have been trying to show: $\sum_{n=0}^\infty {\frac{x^{2n+1}}{(2n+1)!}} = \left(\frac{e^x - e^{-x}}{2} \right)$ I have come so far as to show: $\begin{aligned} \sum_{n=0}^{\infty} {\frac{x^{2n+1}}...
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0answers
21 views

swapping the order of summation [on hold]

https://photos.app.goo.gl/qgVNDcGe9GbRycCa6 [enter link description here][1]
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2answers
25 views

Why is $\sum_{n=0}^{N-1}e^{-2i\pi(k+k_0)n/N}=N\delta(k-N+k_0)$?

EDIT: $\delta$ is the Dirac delta function and in the context it is defined as $\delta(0)=1$ and $0$ for all $n\neq 0$. I am having trouble concluding that $\sum_{n=0}^{N-1}e^{-2i\pi(k+k_0)n/N}=N\...
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0answers
68 views

Approximating an infinite sum: $\sum_t t^d \exp(-(t-1)^2/2)$

I am seeking to upper bound the limit of the following infinite summation, when a free parameter $\beta$ can be chosen, perhaps dependent on $d$, to help reduce the sum: \begin{equation} f(\beta,d) = \...
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2answers
63 views

Sum of the sum of the sum of $4i(-1)^i$

I am attempting to evaluate the sum $$ \sum_{i=0}^{x-1} \Biggl(4+4\sum_{j=3}^{i+2} j(-1)^j\Biggr) $$ I'm not that experienced at summations, so I can go as far as $$ 4x+4\sum_{i=0}^{x-1} \sum_{j=3}^{...
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9 views

Normal Equations (OLS) : Summation notation to system of equation form

I am struggling to "visualize" the following equation. I would like to know how the following equation can be written in system-of-equation form in order to see that there are $K$ unknowns. The ...
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2answers
37 views

Sum $\sum_{i = 1}^N \sum_{j = i + 1}^N \mathbb{I} (j - i \leqslant K)$

Let $N,K$ be non-negative integers. What's the value of the following sum? $$S(N,K) = \sum_{i = 1}^N \sum_{j = i + 1}^N \mathbb{I} (j - i \leqslant K)$$ where $\mathbb{I}(\mathcal P)=1$ if $\mathcal ...
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3answers
40 views

Summation formula for this?

I have found the following summation formula based on a recurrence. It supposes $n = 2^k$ where k is an integer. I've intuitively discovered that the following closed form may be true (following the ...
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1answer
41 views

What constant $c$ will make $ \sum_{k=2}^{N}c^{\frac{1}{k\log k}}=N ?$

What constant $c$ will make this equality valid for any $N$ chosen? $$ \sum_{k=2}^{N}c^{\frac{1}{k\log k}}=N. $$ I tried getting a rough idea of what $c$ should be and got about $1.46$ when $N=1000$ ...
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4answers
212 views

Closed form of recurrent arithmetic series summation

Knowing that $$\sum_{i=1}^n i = \frac{n(n+1)}{2}$$ how can I get closed form formula for $$\sum_{i=1}^n \sum_{j=1}^i j$$ or $$\sum_{i=1}^n \sum_{j=1}^i \sum_{k=1}^j k$$ or any x times neasted ...
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1answer
83 views

Find $\lim_{t\to 1^{-}}(1-t)\sum_{r = 1}^\infty \frac{t^r}{t^r+1}$

$$\lim_{t\to 1^{-}}(1-t)\sum_{r = 1}^\infty \frac{t^r}{t^r+1}$$ Note: I am a high school student and this problem appeared in my test. So, please try to use methods to solve this problem at a high ...
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2answers
48 views

Sum of alternating binomial-coefficient-type series

Let $D,n\in \mathbb N$ with $0<D<n$, and $y>0$ is a real number. Question: Is there a closed-form for the following alternating sequence \begin{equation} \sum_{k=0}^D (-y)^k {n\choose k}? \...
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3answers
327 views

What is the value of $\frac11+\frac13-\frac15-\frac17+\frac19+\frac1{11}-\dots$?

The series $\sum_{k=1}^{\infty }\frac{(-1)^{k+1}}{2k-1}=\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\dots$ converges to $\frac{\pi}{4}$. Here, the sign alternates every term. The series $\sum_{k=...
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1answer
11 views

Algorithm Complexity - Summation - Correctly interpreted how to do it.

I wanted to double check my understanding and working out for a 3 nested for loop algorithm, and working out it's complexity. I've got the right answer, but how I've arrived at it I feel isn't exactly ...
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1answer
61 views

$\sum_{n=1}^\infty \frac{n}{4n^4+1}$ converges to? [closed]

$$\sum_{n=1}^\infty \frac{n}{4n^4+1}$$ my attempt : assumed the series is a telescopic and tried finding $t_n - t_{n-1}$ but then realized it is not a telescopic series. $$$$ //answer is given to be 0....
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0answers
31 views

Separating out the last term of a given sequence

I am suppose to separate the last term of the given sequence. $$\sum_{j=0}^n2^j$$ my work as shown... $$ \sum_{j=0}^{n-1}2^j\ +\ 2^n$$ Does this appear wrong?
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0answers
33 views

Help on converting summation to integration [closed]

Can someone help me on turning the sum $$\sum_{i=1}^\infty 1+\frac{i\lfloor\frac{x}{i}\rfloor-x+\frac{1}{2}}{\vert i\lfloor\frac{x}{i}\rfloor-x+\frac{1}{2}\vert}$$ into an integral, I have tried but ...
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1answer
38 views

Infinite sums and squares

So, I'm sure this has been thought of and said before but I'm curious. So $\sum \frac{1}{2^n}$ can be thought of by filling up a square. First we color in a whole square and then we draw a second ...
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1answer
49 views

How to convert an infinite summation to an integral [closed]

I need help converting this summation to an integral $$\sum_{i=1}^\infty 1+\frac{\vert(n-im)\vert}{(n-im)}$$ I keep trying but get stuck so any help is appreciated.
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0answers
23 views

Rewriting a set of integers to get rid of repetition but keeping subset sum ordering

Say, I have a set of 6 +ve integers sorted in ascending order: $A = \{2,4,4,4,5,7\}$ Now to make it easier to deal with (Minimum one starts with 1) I deducted one from all of them: $\therefore B= ...
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1answer
69 views

Find the infinite sum $\sum^\infty_{n=2}\frac{\left(-1\right)^{n}}{\log n}$, speed up its convergence

$\displaystyle\sum_{n\in\mathbb{Z}_{\geqslant{2}}}\frac{\left(-1\right)^{n}}{\log n}$ Since $\log x$ grows more slowly than any positive power, the sum above converges extremely slowly. Is there a ...
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0answers
14 views

Proving $\sum_{j=0}^{N-1}\cos\frac{\left(2j+1\right)\pi}{2N}=0$ [duplicate]

Let $l\in\mathbb{Z}$ and $N\in\mathbb{N}$. I need to prove the following: \begin{equation} \sum_{j=0}^{N-1}\cos\left(l\frac{\left(2j+1\right)\pi}{2N} \right)=0 \end{equation} I tried to use Euler ...
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1answer
43 views

Find the Sum of the Series: $\sum_{0}^{\infty}\frac{(-1)^n\pi^{2n}}{6^{2n}(2n)!}$

Find the Sum of the Series $$\sum_{n=0}^{\infty}\frac{(-1)^n\pi^{2n}}{6^{2n}(2n)!}$$ Alright, so I think I may have gotten this problem correct but I'm a little hesitant, so If you could check my ...
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2answers
85 views

Derive the sum of $\sum_{i=1}^n ix^{i-1}$

For the series $$1 + 2x + 3x^2 + 4x^3 + 5x^4 + ... + nx^{n-1}+... $$ and $x \ne 1, |x| < 1$. I need to find partial sums and finally, the sum $S_n$ of series. Here is what I've tried: We ...
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0answers
65 views

What is $\sum_{k=1}^m\left[2^k\begin{pmatrix} n \\ k \end{pmatrix}\right]^2$?

The equations $$ \sum_{k=1}^m\begin{pmatrix} n \\ k \end{pmatrix}^2 \quad \text{and} \quad \sum_{k=1}^m\left[2^k\begin{pmatrix} n \\ k \end{pmatrix}\right]^2$$ popped up in some of my calculations, ...
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0answers
25 views

Is it incorrect to have a sum of an infinite weighted set?

I am currently in a revision of a paper. I have found something that I would only like to change if it is currently really notationally false because each change bears a few risks. What I currently ...
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2answers
78 views

Find the partial sum formula of $\sum_{i=1}^n \frac{x^{2^{i-1}}}{1-x^{2^i}}$

Given next series: $$\frac{x}{1 - x^2} + \frac{x^2}{1 - x^4} + \frac{x^4}{1 - x^8} + \frac{x^8}{1 - x^{16}} + \frac{x^{16}}{1 - x^{32}} + ... $$ and $|x| < 1$. Need to derive $S_n$ formula ...
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1answer
42 views

Sum of sequential numbers rational or irrational? [closed]

Is following infinite sum is rational or irrational? $\sum_{n=1}^\infty n/10^n$
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2answers
38 views

$\sum_{n=1}^{\infty}\sum_{i=nr}^{\infty}f(i)=\sum_{i=r}^{\infty}\sum_{n=1}^{\left\lfloor \frac{i}{r}\right\rfloor}f(i)$

This was given as part of the answer of a more complex problem: $$\sum_{n=1}^{\infty}\sum_{i=nr}^{\infty}f(i)=\sum_{i=r}^{\infty}\sum_{n=1}^{\left\lfloor \frac{i}{r}\right\rfloor}f(i)=\sum_{i=r}^{\...
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2answers
48 views

Complex series involving hyperbolic cosine [on hold]

Please how to calculate the sum of such series! I need the idea ! $$\sum _{n=1}^{\infty} \cosh(n)\frac{z^{2n}}{n!} $$ $$\sum _ {n=0}^{\infty} \frac{(1+i)^{n}z^{n}}{n!}$$
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3answers
31 views

Evaluate $\sum_{0}^{\infty}\frac{(-1)^n\pi^{2n+1}}{3^{2n}(2n+1)!}$

Evaluate $\sum_{0}^{\infty}\frac{(-1)^n\pi^{2n+1}}{3^{2n}(2n+1)!}$ For this homework problem, I really don't have any clue how to start it, so any hints are welcome. But my first intuition would be ...
0
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2answers
25 views

Exponential double angle formula

My question is whether someone could provide a proof for the following identity: $$ \frac{1 - e^{int}}{1 - e^{it}} = e^{i(n-1)t/2} \frac{\sin(nt/2)}{\sin(t/2)} $$ Motivation: The left hand side is ...
1
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4answers
54 views

Find the value of $\sum_{r=1}^4 \log_2 (\sin(\frac{r\pi}{5}))$

Find the value of $$\sum_{r=1}^4 \log_2 (\sin(\frac{r\pi}{5}))$$ My apporach:- $$\sum_{r=1}^4 \log_2 (\sin(\frac{r\pi}{5}))$$ $$=\log_2 (\sin(36^{\circ}))+\log_2 (\sin(2*36^{\circ}))+\log_2 (\sin(3*...
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1answer
30 views

Find the Taylor series of $\frac{1+z}{1-z}$ at $z_{0}=i$

I'm given the following explanation: Let $\dfrac{1+z}{1-z} = \biggl(\dfrac{1+i}{1-i}+\dfrac{z-i}{1-i}\biggr)\biggl(1-\dfrac{z-1}{1-i}\biggr)^{-1}$ = = $\dfrac{1+i}{1-i}\displaystyle\sum_{j=0}^{\...
0
votes
2answers
34 views

Find the value of $a$ such that the series converges.

I have to find the value of a such that the following series converges:$$\sum_{n=1}^{\infty}n^{\frac{1}{3}}\left|\sin\left(\frac{1}{n}\right)-\frac{1}{n^a}\right|$$ First of all, I simplified the ...
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votes
0answers
39 views

Decimal digit extraction of $\pi$

I've seen many folks asking for this so I thought I'd take a shot @ answering it: This function $\pi[d]:\mathbb{N}\rightarrow\left\{\mathbb{W}<10\right\}$, based on the BBP closed form expression, ...
2
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2answers
69 views

Evaluating $\sum_{k=1}^{\infty}\frac{1}{k(3k-1)}$

I am wondering if the sum $$S=\sum_{k=1}^{\infty}\frac{1}{k(3k-1)}$$ has an exact expression. And when I plugged it into Wolfram Alpha it spitted out: $$S=\frac{1}{6}\Big(-\sqrt{3} π + 9 \ln(3)\Big)$$...
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3answers
46 views

Why is $\sum\limits_{i = 1}^n \frac{n}{n-i+1}$ equal to $n \sum\limits_{i = 1}^n \frac{1}{i}$?

Assume we have the sum $\sum\limits_{i = 1}^n \frac{n}{n-i+1}$ why is this equal to $n \sum\limits_{i = 1}^n \frac{1}{i}$?