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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Inequality with interesting independent constants

Let $b_1,\dots,b_{n-1}$ be integers satisfying $0 \le b_i \le n-i$ for each $i \in [n-1]$ such that $\sum_{i=1}^{n-1} b_i = \alpha \binom{n}{2}$ where $\alpha$ is constant strictly between $0$ and $1$....
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3 votes
3 answers
89 views

Evaluating $\sum_{k=0}^{n}\binom{n}{k}\binom{k}{r}\binom{k}{s}(-1)^{k}$

So far I've been able to determine that if $n, r, s$ are nonnegative integers, then $$ \sum_{k=0}^{n}\binom{n}{k}\binom{k}{r}\binom{k}{s}(-1)^{k} = \begin{cases} 0 &\qquad\text{ if } r+s < n, \\...
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1 vote
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Evaluate $\sum_{n=1}^\infty (\frac{1}{2}\sin x)^n $

I'm trying to answer this having been sent a photo of the questions from someone doing A level Edexcel Maths and I'm struggling. Any advice would be appreciated. Evaluate and justify the validity of ...
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2 votes
1 answer
29 views

Evaluate/asymptotic of the sum $\sum_{a = 1}^{L \left({N}\right)} \left({- 1}\right)^{\left\lfloor{N/\left({2\, a + 1}\right)}\right\rfloor}$

I am trying to evaluate the sum and its asymptotic limit as $N \rightarrow \infty$ of $$\sum_{a = 1}^{L \left({N}\right)} \left({- 1}\right)^{\left\lfloor{N/\left({2\, a + 1}\right)}\right\rfloor}$$ ...
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0 votes
2 answers
44 views

Transforming a double sum into a product of two sums.

Question is from Stein-Shakarchi Vol. I. I am asked to show that for two complex numbers $z_1$ and $z_2$, $e^{z_1}e^{z_2}=e^{z_1+z_2}$ using the definition of the complex exponential $e^z=\sum_{n=0}^{\...
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7 votes
4 answers
181 views

Inequality involving sums with binomial coefficient

I am trying to show upper- and lower-bounds on $$\frac{1}{2^n}\sum_{i=0}^n\binom{n}{i}\min(i, n-i)$$ (where $n\geq 1$) in order to show that it basically grows as $\Theta(n)$. The upper-bound is easy ...
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6 votes
0 answers
74 views

A curious limit: $\lim\limits_{n\to\infty}\sum\limits_{i=1}^{n}\left[\left(\frac{n}{n+1-i}\right)\right]^{a}f(i) = c\sum\limits_{i\geq 1}f(i)$

I am trying to prove, for the general case whereby $\zeta(\cdot\,,\cdot)$ is the Hurwitz-Zeta function, and $a\in \mathbb{N}$, that $$\mathcal{L} = \lim\limits_{n\to\infty^{+}}\sum\limits_{i=1}^{n}\...
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0 votes
1 answer
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How to bound $\sum e^{-a n^p}$

Let $0<p<1$ and $a>0$. Then it would seem that $$ \sum_1^\infty e^{-an^p}\le Ce^{-a} $$ For some constant $C(p)$ since the terms in the summation decay exponentially. However, I can't quite ...
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Relation between sums

Let $$A=\sum_{i=1}^N a_i\\ B=\sum_{i=1}^N b_i$$ where $0<a_i,b_i<1\ \forall i$ and $N <+\infty$ Let $$a=A/N$$ $$b=B/N$$ such as $\sum_{i=1}^N a=A$ and $\sum^N_{i=1}b=B$ is there any kind of ...
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0 answers
24 views

Decomposition of a summation to a product into smaller summations

I came across a proof that used the fact that $$\sum_{p=1}^{mn}p = \sum _{i=1}^{m}\sum _{j=1}^{n}(n(i-1)+j).$$ In general, I found that $$\sum_{p=1}^{\Pi_{k=1}^{n}{m_k}}p = \sum _{i_1=1}^{m_1}\sum _{...
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0 votes
3 answers
63 views

What is sum of $\sum^{\infty}_{k=3} \frac{q^k}{k}$? [closed]

what is the sum of $\sum^{\infty}_{k=3} \frac{q^k}{k}$, where $q \in (0,1)$?
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0 votes
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Summation: Adding all scores with a min and max (score) of 10 and multiply by (amount of Issues) Y - Notation Formatting Advice

I am new this portal and haven't done any maths for a very long time. For an academic project I am working on, I attempting to create a equation (not even sure that's the right language to use) where ...
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7 votes
3 answers
184 views

How to evaluate the sum of $\sum_{n=0}^{\infty}\frac{1}{3n^{2}+4n+1}$

I hava an infinite sum $$\sum_{n=0}^{\infty}\frac{1}{3n^{2}+4n+1}$$ I factored the denominator $$\sum_{n=0}^{\infty}\frac{1}{\left(3n+1\right)\left(n+1\right)}$$ Then I separated the fraction $$\frac{...
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0 votes
0 answers
58 views

How do I find the partial sum of the Maclaurin series for $e^x$?

In one of the problems I am trying to solve, it basically narrowed down to finding the sum $$\sum^{n=c}_{n=0}\frac{x^n}{n!}$$ which is the partial sum of the Maclaurin series for $e^x$. Wolfram | ...
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  • 1,690
3 votes
1 answer
38 views

Proving by Induction a Combinatorial Binomial

I'm currently stuck trying to prove that for all items in the sequence $\binom{n+1}{2}$, for all $n\geq 1$, that $\binom{n+1}{2}=\sum \limits _{i=0}^ni$. My first assumption is to solve for my base ...
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0 votes
3 answers
54 views

Show that $\sum_{r=1}^{n} \frac{5r+4}{r(r+1)(r+2)}=\frac{7n^2+11n}{2(n+1)(n+2)}$ [closed]

Show that: $$\sum_{r=1}^{n} \frac{5r+4}{r(r+1)(r+2)}=\frac{7n^2+11n}{2(n+1)(n+2)}$$ I'm very lost on this question. I initially tried to identify a certain progression that this function adhered to, ...
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  • 315
0 votes
1 answer
66 views

Maximize $\sum_{i=1}^N \log\left( 1+a_i b_i c_i \right)$ under $\sum_{i=1}^N a_i \to \infty$ and $\sum_{i=1}^N a_i \to 0$

I have a log-sum maximization problem of the form: $$ \max_{\left\{ a_i \right\},\left\{ b_i \right\},\left\{ c_i \right\}} ~ \sum_{i=1}^N \log\left(1+ a_i b_i c_i \right) $$ subject to $$ \sum_{i=1}^...
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0 answers
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Derivation of an upper bound

This is question 24 of this collection of dice problems with solutions, I couldn't understand this $$\begin{align} \frac{1+5^n+\max\{1+6,1+C\log(n-1)\}\sum_{j=1}^{n-1}{n \choose n-j}5^j}{6^n-5^n}=\...
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0 votes
1 answer
30 views

How to show that $\sum_{i=1}^n\left(x_i^2 - \bar{x}^2\right) = \sum_{i=1}^n\left(x_i - \bar{x}\right)^2$

I am reading through a book on linear regression and I am confused as to how a derivation has been done. The derivation up to where I have got is below. $$ \sum_{i=1}^nx_i^2 - \frac{\left(\sum_{i=1}^...
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-1 votes
0 answers
38 views

Simplifying $ \sum_{k=1}^n\binom n{k-1}\frac{x^{k+n}y^{n-k+1}}k. $ [closed]

I want to simplify this binomial expression: $$ \sum_{k=1}^n\binom n{k-1}\frac{x^{k+n}y^{n-k+1}}k. $$ I tried to simplify it but it's pretty hard . So if someone can help with a hint or solution.
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0 votes
1 answer
35 views

What is the general expression for the number of possible way to generate $k$ distinct number from $n$ sequentially?

What is the general expression for the number of possible way to generate $k$ distinct number from $n$ sequentially? For example, let's say I have $n$ numbers ranging from $1$ to $9$ and I want to ...
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  • 1
0 votes
0 answers
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A complicated sum of cosines involving sums inside the cosine arguments

Let $n$ be an integer greater than $1$. How do I show that $$\sum_{L_{1}=0}^{n-2}\sum_{L_{2}=0}^{L_{1}}\left(-1\right)^{L_{1}+L_{2}}\left(1+\left(-1\right)^{L_{1}+L_{2}}\cos\left(\pi\sum_{k=L_{2}+1}^{...
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0 votes
1 answer
26 views

Why is this step required in the proof of sum of first $n$ odd numbers using the Well Ordering Principle?

I came across this question while doing $\text{6.042J}$ from MITOCW. I have a doubt in the part c, namely, why do we need to manipulate the formula in that way? Here is my solution so far to the ...
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  • 1,370
-1 votes
1 answer
60 views

Finite summation including binomial coefficients [closed]

I'm trying to solve the summation below, but I can't get a nice algebraically reduced equation. $$\sum_{k=1}^{2n} (-1)^{k-1}\frac{1}{\binom{2n}{k}} $$ I have tried to convert this into $\sum_{k=1}^{n}\...
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0 votes
1 answer
22 views

Abbreviating lots of consecutive indexed summations

I have lots of equations of the following form $$\sum_{r_0}\sum_{r_1}\sum_{r_2}\cdots\sum_{r_N} x_{r_0} x_{r_1} x_{r_2}\cdots x_{r_N} $$ I can use the following notation for the product of $x$s $$x_{...
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  • 1,407
1 vote
0 answers
58 views

sum of the series $\sum_{n=0}^\infty\frac{2x}{x^2+(2n+1)^2\pi^2}$

Is there a simple way to prove this equality valid for all $x\in \Bbb R$: $$\frac1{1+e^x}=\frac12-\sum_{n=0}^\infty\frac{2x}{x^2+(2n+1)^2\pi^2}.$$ My idea is to use the uniform convergence of this ...
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0 votes
1 answer
25 views

Find list of N increasing values that add up to S given start value V

I have a list of N items The price of the 1st item in the list is valueA I want the price to gradually increase from one item to ...
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0 votes
1 answer
81 views
+50

A sum related to the Mobius function

It is well-known that $$\frac{\phi(n)}{n}=\sum_{d\mid n}\frac{\mu (d)}{d} \quad n\in \mathbb Z^+$$ Where $\phi $ is Euler's totient function and $\mu$ is the Mobius function. But using the formula for ...
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0 answers
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Prove that if $n = \sum_{i=1}^n |y|^i =|\sum_{i=1}^n y^i|$ for a complex number y with $|y|\leq 1$, then $y=1$.

Let $n\in\mathbb{Z}^+.$ Prove or disprove that if $\sum_{j=1}^n |y|^j =|\sum_{j=1}^n y^j|$ for a complex number y with $|y|\leq 1$, then $y=1$. Does the statement follow if we assume $n = \sum_{j=1}^n ...
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0 votes
0 answers
27 views

If there are closed form for $\sum\limits_{m=i}^\infty \frac{2}{(2m-1)(2m-1)!}\binom{2m-1}{m-i}$

If there are closed form for $$\sum\limits_{m=i}^\infty \frac{2}{(2m-1)(2m-1)!}\binom{2m-1}{m-i},$$ where $m,i$ are positive integers. I have tried defining the generating function $$\sum\limits_{i=1}^...
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0 votes
0 answers
34 views

Sum of $ ^nC_k $ terms from $ n $ numbers and each term having $ k $ numbers multiplied .

How can this calculation be done? Imagine I have four consecutive numbers $ 1,2,3,4$. I want to divide this whole set of numbers into $ ^4C_2 $ combinations and then sum up them all. Which is ...
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0 votes
2 answers
44 views

Convergence of series with two variables [closed]

In an electronics application I've found this summation: $$ \sum_{n=1}^{\infty}\sum_{m=0}^{\infty} (-1)^{m - n+1} \cdot \beta^n \cdot \alpha^m $$ Both $|\alpha|$ and $|\beta|$ are <1 . Does this ...
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1 vote
0 answers
38 views

Prove $\sum_{x\le Q} e(\alpha x)\ll \lVert \alpha\rVert^{-1}$ for $\alpha\not \in\mathbb Z$

I just want to know the proof that $\left\vert\sum_{x\le Q} e(\alpha x)\right\vert\ll \lVert \alpha\rVert^{-1}$ for $\alpha\not \in\mathbb Z$, for any positive integer $Q$. Here $\lVert x\rVert$ means ...
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1 vote
1 answer
29 views

Summation Expression with Range and Sequence expression

I've been trying to search for the meaning for this summation for a couple of days now. I've used https://approach0.xyz/ to search on the stack, but I cannot seem to find an explanation that helps me. ...
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-2 votes
0 answers
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A word problem on a sequence of financial values [closed]

I need help on a Series and Sequence Assessment Task. PROBLEM DESCRIPTION A business man in Minnesota sets up a prize fund of $\$3000$ with a single investment to provide an annual prize of $\$200$. ...
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0 votes
1 answer
23 views

Show the rational representation of e "converges to e" under constructive math, i.e. show that $\sum _{k=j+1}^i \frac{1}{k!} \le \frac{1}{n}$.

I'm trying to show that $e$ has a representation in constructive mathematics from Wikipedia's definition of convergence and definition of finite e. Here is their definition of "convergence": ...
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0 votes
1 answer
39 views

Quick formula for $f(n) = \sum_{i=0}^{i=n}{k^{ji}}$ where $k \in \boldsymbol{\mathbb{R}}$ & $j \in \boldsymbol{\mathbb{N}}$

I am trying to figure out a function $f(n)$ which takes the input $n$, where $n \in \boldsymbol{\mathbb{N}}$, and outputs the sum $\sum_{i=0}^{i=n}{k^{ji}}$ till the $n^{th}$ value where $k \in \...
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1 vote
0 answers
16 views

proofs about a norm involving an integral and fourier coefficients

Let $C([-\pi, \pi])'$ be the set of continuous functions $f$ from $[-\pi, \pi]$ to $\mathbb{C}$ such that $f(\pi) = f(-\pi)$. For $f \in C([-\pi, \pi])', n \in \mathbb{Z}$, define $a_n = \frac{1}{2\pi}...
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  • 881
0 votes
1 answer
26 views

Number of possible combinations of X numbers that sum to Y where the order doesn't matters

I am looking for the number of possible outcomes given to a set of numbers X that sum to Y. This is the same issue as here. However, I would like to consider that (i) the numbers can't be repeated and ...
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0 votes
0 answers
21 views

Notation for Expresion of Tangent of Summation of Angles

I am proving, assuming this expression leads to something correct, that the tangent of a summation of angles is an expression involving sums of products of the tangents of each angle, like this: $$ \...
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  • 2,700
0 votes
0 answers
25 views

Computing $R(4n+1,4n+3,4n+5)$, $n=1,3,5,\dots$

For pairwise positive integers $a_1,a_2,a_3$, define $$R(a_1,a_2,a_3)=\frac{1}{a_1a_2a_3}+ \sum_{i=1}^3 \frac{2}{a_i} \sum_{k=1}^{a_i-1} \cot\left(\frac{\pi a_1a_2a_3 k}{a_i^2} \right)\cot\left(\frac{...
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  • 2,034
2 votes
3 answers
54 views

Prove that for $\alpha \leq2$ the sum $\sum_{n=1}^\infty (\sqrt{n+1}-\sqrt{n})^\alpha$ is divergent, and if $\alpha \geq 4$ the sum is convergent.

Prove that for $\alpha \leq2$ the sum $\sum_{n=1}^\infty (\sqrt{n+1}-\sqrt{n})^\alpha$ is divergent, and if $\alpha \geq 4$ the sum is convergent. Attempt: $$\sum_{n=1}^\infty (\sqrt{n+1}-\sqrt{n})^\...
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1 vote
1 answer
47 views

How does one translate summation to matrix multiplication

This is probably a very basic question, but after a day of research I still cannot figure it out. In general my question is as follows: Is there a set of rules how one gets from a summation expression ...
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  • 11
0 votes
0 answers
17 views

formula for finding an upper limit for summation

I am summing over percentages list Per = [5,6,1,4] this list has 4 elemets so I am summing as follows : $\sum_{i=1}^{4} per(i)$ the sum is 16 and the half of this sum is 8 $1/2\sum_{i=1}^{4} per(i)$ ...
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  • 25
0 votes
1 answer
44 views

How to calculate this partial sum?

Why is $\displaystyle\sum_{n=0}^{k} \frac{\sin(n \pi)\cos(n \pi t) \sin(n \pi x)}{\pi-\pi n^2}=1/2 \cos(\pi t)\sin(\pi x)?$ How would I calculate this and why is there no $k$ left in the partial sum? ...
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1 vote
1 answer
69 views

How to factorize and solve equations with $\Sigma$ notation?

I have a few doubts about the properties of sigma notation, $\Sigma$ . My questions rely on factorization and solving equations with $\Sigma$.On account of the fact that my questions are correlated, I ...
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1 vote
2 answers
91 views

Summation of roots [closed]

In this post, Calculate summation of square roots we are shown how to sum square roots. My question is, can we get similarly simple expressions if instead of square roots we choose some other exponent....
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  • 321
0 votes
0 answers
25 views

Frobenius norm of sums of matrix products

If we have a sum of products of two conformable matrices: $\sum_{i=1}^NA_i'B_i$, I would like to understand if the following is generally true: $$ \left\|\sum_{i=1}^NA_i'B_i \right\|\leq \max_{1\leq i\...
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  • 11
1 vote
1 answer
57 views

Prove or disprove that $\sum_{k=1}^p G(\lambda^k) = ps(p)$

Prove or disprove the following: if $\lambda$ is a pth root of unity not equal to one, $G(x) = (1+x)(1+x^2)\cdots (1+x^p),$ and $s(p)$ is the sum of the coefficients of $x^n$ for $n$ divisible by $p$ ...
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  • 881
-1 votes
0 answers
26 views

Find a compact form of a given sum

I have a given sum: $S = \sum_{k=1}^{n+1}\sum_{l = 1}^{k}k$ I have to find a compact form of a given sum. I tried that way: $S = \sum_{k=1}^{n+1}\sum_{l = 1}^{k}k = \sum_{k=1}^{n+1}k^2 = 1^2 + 2^2 + ...
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