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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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2answers
33 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
34 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
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0answers
19 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)^{...
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0answers
19 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
26 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 ...
-1
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1answer
15 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
59 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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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?
0
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1answer
27 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
41 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
47 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
23 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{...
0
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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
128 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 ...
0
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1answer
60 views

How to solve this sum $\sum_{n=0}^{\infty} e^{-0.5n(n+1)}$ [on hold]

For my homework I need to solve the sum $$\sum_{n=0}^{\infty} e^{-0.5n(n+1)}$$ I tried to rewrite it as a geometric series because it kind of looks like one but to no success.
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0answers
46 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_{...
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0answers
21 views

Convergence acceleration of a series by using transformation

One of the ways of accelerating the convergence of a series is by transforming into a faster series. Examples of this approach can be found in this paper. A generalization of this method leads to the ...
0
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2answers
45 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
60 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
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1answer
19 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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0answers
21 views

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

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
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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
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0answers
52 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
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1answer
33 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 ...
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1answer
29 views

Summation including combination [on hold]

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
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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
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0answers
15 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 ...
1
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1answer
33 views

are $\sum_{i=0}^{n}i$ and $\sum_{i=n}^{0}i$ equivalent?

So here's the ugly history of how I came to ask this question. I was following this proof: and got stuck at this step: $$\sum_{j=0}^{(\log_2n) - 1}\frac{1}{(\log_2n) - j} = \sum_{l = 1}^{\log_2n}\...
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0answers
26 views

Quadratic martingale bound

I know that if $a_1,a_2,\dots$ are random variables and {$\mathcal{F}_{t}$ } is a filtration such that $$\mathbb{E}[a_i \mid \mathcal{F}_{i-1}] \leq K$$ for all $i$, then for any stopping time $\...
1
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2answers
43 views

Floor function summation[difficult]

The question is to find the value of — $$\sum_{r=1}^{502} \Big \lfloor \frac{305r}{503}\Big \rfloor$$ The answer is pretty big, so I don't think trial and error will work here. I seriously can't ...
1
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1answer
27 views

Pattern in Squared Numbers and their Digit Sum

So this has been boggling my mind for some time now. On spring break I was really bored and started messing around with numbers when I noticed something. I was squaring each number (1-9) when I ...
0
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3answers
63 views

Sum to infinity series

Consider the following series $$\sum_{k=0}^{\infty}\frac{(k+1)(k+3)(-1)^k}{3^k}$$ I'm told to analytically find the sum to infinity and I have been given this as a clue. $$\Sigma_{k=0}^\infty x^k = ...
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2answers
164 views
+50

Evaluating sum $\sum_{m=0}^{\infty}\frac{(2-\delta_m^0)(-1)^m \lambda_0}{a(\lambda_0^2 -(\frac{m\pi}{a}))}\cos(m\pi x/a)$

How Can I evaluate the following sum$$\sum_{m=0}^{\infty}\frac{2-\delta_m^0}{a}\frac{(-1)^m \lambda_0}{\lambda_0^2 -(\frac{m\pi}{a})}\cos\left(\frac{m\pi x}{a}\right)=\frac{\cos(\lambda_0 x)}{\sin(\...
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0answers
11 views

Stochastic Proof Piece

As a part of the proof of recurrence criterion, (Page 22 here: http://web.math.ku.dk/noter/filer/stoknoter.pdf), it is shown that $(P^n)_{i,j} = \sum_{m=1}^n (P^{n-m})_{j,j}f_{ij}^{(m)}$. This makes ...
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2answers
45 views

How is $\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}x^{i+j} = \sum_{n=0}^{\infty}\sum_{i=0}^{n}x^n$

My professor used this in class for a proof and I'm having trouble understanding it. $\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}x^{i+j} = \sum_{n=0}^{\infty}\sum_{i=0}^{n}x^n$ The way he explained it, ...
-1
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1answer
24 views

Differential Geometry - vector fields Lie bracket

Can anyone tell me why (in the last line) $i$ changes to $j$ in the first component of the sum?
1
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1answer
33 views

Sigma notation with equality conditional

I am used to the Sigma notation: \begin{align} \sum_{j=1}^{n} \frac{1}{j} \end{align} I have the following problem to solve: Let $H_{k}$: \begin{align} \sum\limits_{1 \le j \le k} \frac{1}{j} & ...
1
vote
1answer
19 views

Splitting / rearranging sigma sums

I am struggling to understand the concept of sigma sum rearrangement. In fact, I don't even know what to call it. That being said, if anybody can recommend sources for me to study this from or let me ...
1
vote
1answer
33 views

Values of some “natural” sums over multiindices with a given absolut value

I'd like to know if there is a nice closed expression in terms of $j$ and $k$ of the sum $$ S_{j,k}:=\sum_{(i_1,\ldots,i_k)\in \mathbb{N}^k_0:\\i_1+\cdots+i_k=j}\frac{1}{i_1!\cdots i_k!}. $$ ...
0
votes
1answer
20 views

Show that $\sum^{N-1}_{i=1} \frac{N-1}{i(N-i)}$ is approximately $2\log N$ for large $N$.

I am working on a problem relating to a Markov chain, that results in the sum: $$\sum^{N-1}_{i=1} \frac{N-1}{i(N-i)}$$ For the expected time until reaching the $N^{th}$ state from $0$. The ...
1
vote
1answer
46 views

A Tricky sum to evaluate (Haldane)

I'm trying to find a way to evaluate this sum (found by Haldane in Phys. Rev. Lett. 60, 635 (1988): $$S_{pq}=\sum_{n=1}^{N-1} z^{nJ} (1-z^{n})^{p-1}(1-z^{-n})^{q-1}$$ with $z= e^{\frac{2i\pi}{N}}$ ...
0
votes
1answer
22 views

Differential of sum of logs of probabilities

I'm trying to calculate the derivative with respect to $p1$, $p2$, ... $pk$ (each) of the following equation: $L = N_1 \log p_1 + N_2 \log p_2 + ... N_k \log p_k$ where $\Sigma_{i=1}^k p_i = 1$ i.e....
0
votes
0answers
21 views

Derivative of a sum

$$L(a,b) = {\sum_{i=1}^n (y^i - (ax^i +b)^2)^2})$$ $$\frac{dL}{da} = \sum_{i=1}^n \frac{(y^i-y^-)*(x^i-x^-))}{var(x)}$$ How can I canclulate the $\frac{dL}{da}$? $$2(y^i - (ax^i +b))^2)* \frac{d(y^i ...