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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.

3
votes
2answers
101 views

If $x^n-3x^{n-1}+2x+1=0$, compute $\sum_{k=1}^n \frac{x_k}{x_k-1}$ [closed]

Let $x_1, x_2, \ldots, x_n$ be the $n$ roots of the equation $x^n-3x^{n-1}+2x+1=0$. Calculate the sum $\sum_{k=1}^n \frac{x_k}{x_k-1}$. Can somebody give me some tips, please? It's pretty hard to ...
0
votes
2answers
53 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 ...
4
votes
1answer
38 views

Value of a binomial series

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. ...
0
votes
0answers
36 views

What are the elements of a fractional summation?

For example, what are the elements in: $$\sum_{i=0}^{2.5} i$$ Background: As part of a program I wanted to generate a sequence along $i^2$ in a range $[a,b]$ with a given sum $c$. So, \begin{align}...
1
vote
1answer
74 views

Proof idea for sum of square / square of sum inequality

This problem came up in a long proof and it is not clear to me how to show this. I tried to apply Cauchy-Schwarz but it is not tight enough. Any idea? Let $\Delta$ be a matrix of real values with $B$ ...
5
votes
1answer
379 views

A curious summation question from pre-RMO 2016

This problem came in the Pre-RMO (Regional Mathematics Olympiad) - Delhi Region (INDIA). I've been solving these questions, and this particular summation question is proving to be quite difficult. ...
1
vote
2answers
35 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
0answers
22 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)^{...
8
votes
2answers
166 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(\...
1
vote
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 ...
-3
votes
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
vote
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}^\...
-1
votes
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 ...
-4
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?
0
votes
3answers
42 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 ...
0
votes
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 ...
2
votes
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, ...
1
vote
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
votes
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
votes
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
votes
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 ...
1
vote
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} ...
14
votes
8answers
1k views

Truncated alternating binomial sum

It is easily checked that $\displaystyle\sum_{i\ =\ 0}^{n}\left(\, -1\,\right)^{i} \binom{n}{i} = 0$, for example by appealing to the binomial theorem. I'm trying to figure out what happens with the ...
13
votes
3answers
408 views

Integral $\int^1_0\frac{\ln{x} \ \mathrm{Li}_2(x)}{1-x}dx$

I would like to know how to evaluate the integral $$\int^1_0\frac{\ln{x} \ \mathrm{Li}_2(x)}{1-x}dx$$ I tried expanding the integrand as a series but made little progress as I do not know how to ...
2
votes
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}\...
0
votes
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 $\...
3
votes
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
votes
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?) $\sum_{k} A_{a_{1},...,a_{i},k}*B_{k,b_{1},...,b_{j}}$ ...
0
votes
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.
0
votes
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
votes
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_{...
1
vote
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
votes
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\...
2
votes
2answers
9k views

Infinitely Many Circles in an Equilateral Triangle

In the figure there are infinitely many circles approaching the vertices of an equilateral triangle, each circle touching other circles and sides of the triangle. If the triangle has sides of length 1,...
1
vote
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 ...
1
vote
1answer
230 views

Finding $\binom{999}{0}-\binom{999}{2}+\binom{999}{4}-\binom{999}{6}+\cdots +\binom{999}{996}-\binom{999}{998}$

Find $$\binom{999}{0}-\binom{999}{2}+\binom{999}{4}-\binom{999}{6}+\cdots +\binom{999}{996}-\binom{999}{998}$$ A.$-2^{500}$ B.$-2^{499}$ C.$2^{500}$ D.$2^{499}$ By the way, I want to ask is there ...
0
votes
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\...
-1
votes
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 ...
1
vote
0answers
27 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 $\...
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
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 ...
0
votes
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 ...
-1
votes
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
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
2answers
110 views

Compute $(\sum_{i=1}^5 x_i)+10$ with given $x_i$'s

So I need help computing the following sum. Question: Suppose $ x_1 = 5, x_2 = −3, x_3 = 7, x_4 = 8,$ and $x_5 = 2$. Compute the following sum. $(\sum_{i=1}^5 x_i)+10$ where the upper limit is 5, ...
1
vote
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
vote
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}\...
0
votes
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
vote
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
votes
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 = ...