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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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1answer
36 views

Calculate limit in use of integrals.

Calculate limit in use of integrals: $$ \lim_{n \rightarrow \infty} \sum_{k=1}^{n} \frac{k+n}{3k^2+n^2+1} $$ Solution: $$\sum_{k=1}^{n} \frac{k+n}{3k^2+n^2+1} = \frac{1}{n} \sum_{k=1}^{n} \frac{\...
2
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2answers
32 views

Calculate limit in use of integrals

Calculate limit in use of integrals $$ \lim_{n \rightarrow \infty} \sum_{k=1}^{n} \frac{1+n}{3k^2+n^2} $$ My attempt: $$\sum_{k=1}^{n} \frac{1+n}{3k^2+n^2} = \frac{1}{n} \sum_{k=1}^{n} \frac{\frac{...
2
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1answer
26 views

Prove identity $\sum^n_{k = 0} \binom {r+k} {r} = \binom{r+n+1} {r+1}$ using lattice paths

I am trying to prove the following identity $\sum^n_{k = 0} \binom {r+k} {r} = \binom{r+n+1} {r+1}$ by using lattice paths. My first approach was to draw the following scheme: Sketch indicating paths ...
3
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0answers
35 views

Products of trig functions and the Thue–Morse sequence

I was studying transformations of finite products of trig functions into sums, and empirically observed that the following curious identity appears to hold for all non-negative integer $m$: $$\prod_{...
0
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3answers
59 views

Find $S_{n}=\sum_{k=1}^{n}k!(k^2+1)$

$$n\in\mathbb{N}^{*}; S_{n}=\sum_{k=1}^{n}k!(k^2+1)$$ I need to find $S_n$ I started like this: $S_{n}=\sum_{k=1}^{n}(k+2)!-3(k+1)!+2k!$ How to continue?I tried to give the k values but the terms ...
0
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2answers
37 views

$S=\sum_{k=2}^{n}\frac{k^{2}-2}{k!}, n\geq 2$

$$S=\sum_{k=2}^{n}\frac{k^{2}-2}{k!}, n\geq 2$$ I got $S=\sum_{k=2}^{n}\frac{1}{(k-2)!}+\frac{1}{(k-1)!}-\frac{1}{k!}-\frac{1}{k!}$ I give k values but not all terms are vanishing.I remain with $\...
3
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0answers
46 views

Closed form for sum of powers of unit fractions

I wonder if it is possible to evaluate explicitly the sum $$S(N):=\sum_{j=1}^{\left\lfloor\frac{N-1}{2}\right\rfloor}\left(1-\frac{2j}{N}\right)^{N+1},\quad N\in\mathbb{N}.$$ In the large $N$ limit ...
1
vote
1answer
27 views

Change the order of minimum in sums

Is it possible to change the order of minimum as i did below: $$\min_w \sum_x\sum_y f(x,y,w)^2 = \sum_x \min_w \sum_y f(x,y,w)^2 $$ If for all $x$ we have $$\min_w \sum_y f(x,y,w)^2,$$ then could I ...
0
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0answers
67 views

sum of the series: $\cos^3 \alpha +\cos^3 {3\alpha} + \cos^3 {5\alpha}+…+\cos^3 {(2n-1)\alpha}$. [duplicate]

Question: Find the sum of the series: $$\cos^3 \alpha +\cos^3 {3\alpha} + \cos^3 {5\alpha}+\ldots+\cos^3 {(2n-1)\alpha}$$ The book from which this question was taken says that the answer is $$\...
1
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1answer
33 views

$\sum_{y=-\infty}^\infty E[X|Y=y]Pr[Y=y] $ algebraic manipulation

I am trying to understand the proof of $E[E[X|Y]]=E[X]$ and there is one part that I am not getting. $$\sum_{y=-\infty}^\infty E[X|Y=y]Pr[Y=y] $$ I know that the left side by definition equals to ...
0
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1answer
42 views

What is the probability distribution for the number of N-sided die rolls needed to get M unique results?

Suppose you have a fair $N$-sided die. You decide to roll it until $M$ unique values have been produced (i.e. you re-roll all previously rolled values). How many times will you roll the die? (Given $2 ...
3
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2answers
37 views

Prove a sum of sums equals n choose k

In some research I'm doing, I've come across some coefficients I'm calling $\alpha^{n}_{j}$, where $$ \alpha^{n}_{j} = \sum_{k_1 = 1}^{n} \sum_{k_2 = 1}^{n-k_1} ... \sum_{k_j = 1}^{n - k_1 - k_2 -... -...
1
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1answer
43 views

lower bounding logarithm of sums

Is it true that $$ \log\left(\sum_{i=1}^{n} \alpha_i\right) = \log\left(n \frac{1}{n}\sum_{i=1}^{n}\alpha_i\right) = \log(n) + \log\left(\frac{1}{n}\sum_{i=1}^{n} \alpha_i\right) \\ \geq \log(n) + \...
0
votes
1answer
26 views

Why when finding a Laurent series can we factor the function

Say we are trying to find the Laurent series (at zero) of $\frac{cos(z)}{z^{3}}$ why is it we can factor out $z^{-3}$ and multiply the Taylor series expansion of $cos(z)$ at zero when we are trying to ...
2
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0answers
101 views

Show that $\sum x^p$ over primes must have a non-trivial zero.

The sum $\sum x^n$ is unbounded in $|x| \le 1$. Similarly if $p$ is prime then trivially $\sum x^p$ is also unbounded in $|x| < 1$ because all primes $> 2$ are odd so the lower bound would ...
1
vote
1answer
35 views

Inequality of double sum with max operator and squared

I am trying to understand a proof, however one inequality remains unclear: $\mathbb{E} \big( \frac{1}{k_1 k_0} \sum_{i=1}^{k_1} \sum_{j=1}^{k_0} a_{j,i} -b_{j,i} \big)^2 \leq \max_i \frac{1}{k_m^2}\...
2
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1answer
47 views

Summation with 3 binomial coefficients [on hold]

In the book "Concrete Mathematics" by Knuth there is a binomial coefficient identity $\sum_{k} {a+b \choose a+k}{b+c \choose b+k}{c+a \choose c+k}(-1)^k=\frac{(a+b+c)!}{a!b!c!}$ but no proof is ...
5
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3answers
130 views

Value of the binomial series $\sum_{i=0}^k \frac{{2i \choose i}}{4^i}$

Some time ago a question was asked here regarding the value of the sum $$\sum_{i=0}^k \frac{{2i \choose i}}{4^i}$$. But it was deleted later by the OP. I went around it but didn't find a solution. ...
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2answers
79 views

Check of $f(x)=\sum_{n=1}^{\infty}\frac{1}{x^2+n^2}$ properties

For function defined as $$ f(x)=\sum_{n=1}^{\infty}\frac{1}{x^2+n^2} $$ check if $f$ is continuous and differentiable function. My approach: I would like to use the connection between this sum and ...
1
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2answers
38 views

Is there a sum of “n+j” terms equivalent to n!

Is there any function so that... $$\sum_{k=0}^{n} f(k) = n!$$ or, $$\sum_{k=0}^{n+j} f(k) = n!$$ where j is any arbitrary integer?
0
votes
1answer
58 views

Double harmonic series $\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\frac{H_{n+m}^{(p)}}{(n+1)^{q}(m+1)^{r}}$

Do these sums exist in the literature and have been investigated before? The same question for the odd variant, that is $$ \sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\frac{O_{n+m}^{(p)}}{(2n+1)^{q}(2m+1)^{...
1
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0answers
20 views

Calculate ssgsea score by hand

I hope this is the right place for this question, if not, feel free to suggest more suitable sites. I would like to calculate a single sample gene set enrichment analysis (ssGSEA) score step by step ...
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0answers
27 views

Sum consecutive tuples [on hold]

I have a set of tuples that can be consecutive based on the last element, like so: (1,2),(2,4),(4,3),(3,2) How can I make the sum of the values without counting the consecutive elements twice and ...
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1answer
16 views

Does multiplying a constant to a summation only apply to the first term?

I used Symbolab to confirm one of my solutions and realized something strange when solving this summation: image of question Symbolab factors out the unneeded constant out of the equation, which is ...
1
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1answer
61 views

Find the coefficient of the power series $[x^3](1-x)^{-1}(1-2x)^6$

I need to find $[x^3](1-x)^{-1}(1-2x)^6$, where $[x^3]$ means the coefficent of the $[x^3]$ term. here's what I've done: $[x^3](1-x)^{-1}(1-2x)^6=[x^3](\sum_{k=0}^6 {6\choose k}(-2x)^k)(\sum_{m=0}^\...
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votes
2answers
34 views

Determine the series whether convergence or divergence with using ratio rest. [on hold]

This is the problem: $$\sum_{n=0}^\infty 3^n\sin((\frac{1}{4})^n)$$ I can't prove the convergence of this series, how can we solve it?
1
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1answer
30 views

How to manipulate this sum-product expression?

A machine randomly outputs either $1$ or $2$, each output being equally likely, and after each output we see the current sum on a screen. What is the probability that a given number $n$ will be ...
1
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3answers
33 views

Sum over the field $\mathbb{F}_{2}^{n}$

Consider the binary field $\mathbb{F}_2$ and then consider $n$ direct products of this: $\mathbb{F}_2 \times \mathbb{F}_2 \times \cdots \times \mathbb{F}_{2}$. Hence, $\mathbb{F}_{2}^{n} = {\{x = (...
2
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1answer
30 views

Fourier transform on $\mathbb{Z}_{2}^{d}$

Let $\mathbb{Z}_{2}^{d} = {\{\textbf{t} = (t_1, \ldots, t_d) : t_j \in \mathbb{Z}_2}\}$. Define the inner product on functions $f, g : \mathbb{Z}_{2}^{d} \rightarrow \mathbb{C}$ to be: $$\langle f, ...
0
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3answers
44 views

How to prove the double sum of combinations is $3^n$

I have a double sum of combinations as follow $$S = \sum_{i=0}^{n}\sum_{k=i}^{n}{n \choose k}{k \choose i}.$$ I guessed and tested that $S = 3^n$, but I have no idea how to prove this. Any help is ...
2
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1answer
50 views

Proof that $f$ is differentiable

Let $$f(x) = \sum_{n=1}^{\infty} \frac{x^n}{2^n} \cos{nx}$$ Proof that $f$ is differentiable on $(-2,2)$ my approach let $ m := \frac{x}{2} $ so $m<1$ $$ \left| \frac{x^n}{2^n} \cos{nx} \right| ...
0
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1answer
25 views

The Lagrange Interpolation formula – Spivak's Calculus Ch 3 Problem 7(b)

The problem: Now find a polynomial function $f$ of degree $n - 1$ such that $f(x_i) = a_i$, where $a, \ldots, a_n$ are given numbers. I found that this question had been asked before, but I did not ...
2
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1answer
45 views

Find the Exact value of $\lim_{n\to\infty} \sum_{k=1}^n \int_0^{\pi} e^{-nx}\sin (kx) dx$

Find the Exact value of : $$\lim_{n\to\infty} \sum_{k=1}^n \int_0^{\pi} e^{-nx}\sin (kx) dx$$ What I did : Take $I=\int_0^{\pi}e^{-nx}\sin(kx)dx$ After Integration, I get $$I=\frac{k(1-e^{-n\pi}\...
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3answers
33 views

Binomial Theorem Inductive Proof - a reindexing moment

I'm copying a proof from someone else and they make this move I don't feel comfortable with. So in the inductive step we assume $ { \left( x+y \right) }^{ n }= \sum _{ m=0 }^{ n }{ \left( \begin{...
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0answers
32 views

Dealing with a polynomial sum

I am trying to approximate the following function, $$f(x)=\frac{\sum^{N}_{c=1}\gamma_c x^{c+k-2}-\sum^{N}_{c=1}\beta_c x^{c-1}}{\sum^{N}_{c=1}\alpha_c x^{c}}$$ where $\alpha_c$'s, $\beta_c$'s and $\...
0
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1answer
12 views

Most elegant way to show Inner-Product of tensors with indices?

How would you write this sum most elegantly? (Or in other words: What is the most elegant and best way to show a sum of products with indicies?) $\sum_{k} A_{a_{1},...,a_{i},k}*B_{k,b_{1},...,b_{j}}$ ...
3
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2answers
131 views

Double Binomial coefficient sum identity $S = \sum_{k = 0}^{n} (-1)^k {{4n-1-2k}\choose{2n-2k}}{{2n-1}\choose{k}}$

I have a sum of factorials that I managed to put in the following form $$S = \sum_{k = 0}^{n} (-1)^k {{4n-1-2k}\choose{2n-2k}}{{2n-1}\choose{k}}$$ where $n\in\mathbb{N}$. Mathematica can sum this ...
1
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0answers
48 views

Converting an Arithmetic Series to Sigma Notation

I've been struggling with the following problem for quite a while now, and have been unable to identify a pattern; You have a geometric series $Y$ for which we have the following rule: $$Y_{t+1} ...
0
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1answer
23 views

Why does the summation of these indicator variables start from i<j?

I'm currently reading through the eighth edition of A First Course in PROBABILITY, by Sheldon Ross. The section I'm reading is "Momens of the number of events that occur", and I understand everything ...
3
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0answers
68 views

Is there any closed expression for $\sum_{k=0}^\infty r^{k^2}$?

Let $r$ be any real number with $0 < r < 1$. Then, of course, there is a closed expression for $$\sum_{k=0}^\infty r^k = \frac{1}{1-r}$$ But does there also exist a closed expression for $$\sum_{...
0
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2answers
46 views

Infinite Sum of Series

So I was given this question $$T_n = \sum _ {k=0}^{ n-1} \frac{n}{n^2+kn+ k^2} $$ And $$S_n = \sum _{ k=1}^n \frac{n}{n^2+kn+ k^2} $$ We were asked wether $T_n$ or$S_n$is$ \gt$or$ \lt \frac{π}{3\...
1
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2answers
61 views

How to interpret an “if and only if” (“iff”) statement in a summation?

I'm a programmer trying to convert the formula below into code, and I don't understand what exactly the "iff" clause on the right side of the numerator is being applied to. The fact that it ...
0
votes
1answer
20 views

Help me to understand the summation notation

I am having following expression, limits of summations are given in terms of set, can someone help me to interpret this summation. where $d_{1}$ and $d_{2}$ are distances. $$\sum_{i_1,i_2 \in \{1,2\...
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votes
0answers
21 views

Trying to evaluate this sum : $\sum_{i=1}^ni^k(n-i)^{r-k}$ [closed]

After long calculations in an exercice I end up with this sum and I wonder if you can simplify it : $$\sum_{i=1}^ni^k(n-i)^{r-k}$$ where $r$, $n$, $k$ are integers, $r\geq k$. I don't really know ...
0
votes
2answers
49 views

Exact sum of the series $\sum_{n=2}^\infty \frac{(-1)^n}{n^3-n}$

I need to find the exact sum of the following series, $\sum_{n=2}^\infty \frac{(-1)^n}{n^3-n}$. The solution goes like this: $\sum_{n=2}^\infty \frac{(-1)^n}{n^3-n}$ $= \frac12\sum_{n=2}^\infty \...
0
votes
0answers
53 views

Calculate the sum.

There is given positive decreasing sequence of real numbers $a_{1},a_{2},a_{3},...$ Calculate the sum in nice form: $$SUM=\sum_{i=1}^{N}a_{i}-2\sum_{1\le i<j\le N}a_{i}a_{j}$$ If it is impossible ...
0
votes
1answer
34 views

Deriving simple formula for summation

I am first year non-math student and I am trying to formalize my homework from digital circuits course using some basic math tools. To give some initial context: If I have $\frac{1}{5}$ frequency ...
-1
votes
1answer
31 views

Summation including combination [closed]

I am having trouble evaluating the summation: $\sum_{k=0}^{2n}(-1)^kk^n{2n \choose k}$ Can anyone lead me to a solution? Also, is there a general or easy way to approach summations that include ...
0
votes
2answers
50 views

Evaluating $\sum\frac{\sin(n)}{n^a}$

I have a function defined as: $$S(a)=\sum_{n=1}^\infty\frac{\sin(n)}{n^a}$$ My question is for what values of $a$ is this convergent, and how can I evaulate this? For starters, I know that $S(a)$ is ...
0
votes
0answers
16 views

How to bound $\sum_m e^{\mathcal{O}(k2^{m/2})-2^{3m/4}}$

I have to estimate the following sum $$ \sum_{m=0}^{\frac{2}{\log 2}\log\log\log T}e^{\mathcal{O}(k2^{m/2})-2^{3m/4}}. $$ I would like to show that this sum is $$ \ll_k 1 $$ and if possible that it ...