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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.

0
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0answers
17 views

Is this sum analytic: $\sum_{\left\{ n_{i}\right\} }\prod_{i}\left(\frac{n_{i}}{\sum_{j}n_{j}}\right)^{M_{i}}\binom{N_{i}}{n_{i}}x^{n_{i}}$?

Let $N_1,\dots,N_K$ and $M_1,\dots,M_K$ be given non-negative integers. Does the following sum admit some analytical simplification? $$\sum_{\left\{ n_{i}\right\} }\prod_{i}\left(\frac{n_{i}}{\sum_{j}...
0
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1answer
29 views

How are these cross-product summations equivalent?

Trying to determine how the $X_{i+1}$ is no longer applicable by changing summation bounds: $$\sum_{i=0}^{n-1} (X_{i} + X_{i+1})(Y_{i+1} - Y_{i}) = \sum_{i=1}^{n} X_{i}(Y_{i+1} - Y_{i - 1})$$ Can ...
0
votes
1answer
53 views

Binomial theorem identity proof $\sum_{r=0}^n \binom{2n}r = 2^{2n-1} + \frac{(2n)!}{2(n!)^2}$ [on hold]

I got this proof recently from a text and I have been at it for 3 hours and I am on the brink of crying lol. Any help ? Number 17 is it By substituting $x=1$ into expansion of $(1+2x)^{2n}$, prove ...
2
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0answers
38 views

A sum that includes power of binomials: Possible limit?

I have the following sum: $$ \sum_{k=1}^{V} (-1)^{k-1} \frac{{V \choose k}^A}{{DV \choose k}^{A-1}} $$ Is there an approximation available for this sum? I computed this sum with python for different ...
1
vote
4answers
66 views

Finding closed form for $\sum_{k=1}^n k2^{k-1}$

I am trying to use the perturbation method to find a closed form for: $$ S_n = \sum_{k=1}^n k2^{k-1} $$ This is what I’ve tried so far: $$ S_n + (n+1)2^n = 1 + \sum_{k=2}^{n+1} k2^{k-1} $$ $$ S_n +...
9
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0answers
108 views

Binomial Coefficients involving Prime Powers Minus 1

I would like to show the following is true; Let $p\in\mathbb{P}, \alpha,n\in\mathbb{N}$. For $$p^\alpha\mid\sum_{k=1}^{n-1}\binom{(p^\alpha-1)n}{(p^\alpha -1)k}$$ I've never worked with prime ...
-1
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2answers
10 views

Summing a series

How would I go about finding the sum of an alternating series that consist of 1 -1/2 + 1/3 -1/4 + 1/5... to the 8th term using a mathematical formula or is there no formula I can use
3
votes
2answers
47 views

Why does this sum converge: $\sum_{k=2}^{n-1} \frac{1}{k} \frac{n \mod k}{n}$

I was playing around with numbers and wanted to create a function that somehow indicates if a number could be a prime. So I came up with this, with the intention that it should make small jumps if $x$ ...
1
vote
3answers
65 views

Sum of sequence $\frac{1}{2} + \frac{3}{4} + \frac{5}{8} + … + \frac{2n - 1}{2^n}$ (not limit for sum) [duplicate]

Can anyone help me to find formula to sum of $n$ first elements in sequence : $$a_n = \frac{2n - 1}{2^n} $$ i.e. : $$S_n = a_1 + a_2 + ... + a_n $$ So can you explain how to find formula for $S_n$...
-2
votes
1answer
22 views

I need help simplifying a sum problem that involves a binomial raised to a power

I have come across a problem in my homework that describes the sum of a binomial squared, and I can't think of a way to simplify it. I have an idea that it would involve $\frac{\left(n\right)\left(n+1\...
0
votes
2answers
58 views

How to get a closed form for $\sum_{i=1}^n \frac{3^i}{2^i}$ [duplicate]

How can I get a closed form for: $$\sum_{i=1}^n \frac{3^i}{2^i}$$ I have just started studying closed forms for summations and I am still lost on this matter so I would appreciate if you guys could ...
0
votes
1answer
30 views

Formula to calculate the sum of combinations [on hold]

I want the formula to calculate the sum of following pattern given n and m: $$^nC_1 * ^mC_1$$ $$+^nC_1 * ^mC_2$$ $$...$$ $$...
0
votes
1answer
73 views

Find a closed-form expression for $\sum_{k=0}^{K} \frac{1}{k+1} {{N-1-k}\choose{K-k}}$ [on hold]

I am trying to find a closed-form expression for the following sum: \begin{align} \sum_{k=0}^{K} \frac{1}{k+1} {{N-1-k}\choose{K-k}} \end{align}
2
votes
1answer
80 views

Evaluating $\lim_{n\to \infty}\sum_{i=1}^n \frac{1}{(i+1)(i+2)}$ through Riemann sum

I am stuck in this problem: $$\lim_{n\to \infty}\sum_{i=1}^n \frac{1}{(i+1)(i+2)}$$ While it can be easily solved using partial fractions, I wanted to solve this through Riemann sums which I terribly ...
1
vote
1answer
67 views

Evaluate the triple sum $\sum_{i=0}^{n-1}\sum_{j=i}^{n-1}\sum_{k=i}^{j} 1$

I have trouble evaluating this triple sum , so far I tried this : $$\sum_{i=0}^{n-1}\sum_{j=i}^{n-1}\sum_{k=i}^{j} 1 = \sum_{i=0}^{n-1}\sum_{j=i}^{n-1}(\sum_{k=i}^{j} 1) $$ $$ = \sum_{i=0}^{n-1}\sum_{...
0
votes
1answer
22 views

Lower bound on Sum of squared differences

Consider $n+2$ real numbers $x_i$ with $0 \leq x_i \leq \frac{1}{2}$. Additionally, not all $x_i$ are the same. Now define two quantities $\Phi = 4\sum_{i=0}^{n+1}x_i^2$ $\Phi' = \sum_{i=1}^{n}(x_{...
0
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2answers
26 views

show that $\sum^\infty_{i=1}p^2(1-p)^{2(i-1)}=\frac{p^2}{1-(1-p)^2}$

In my homework assignment it says: $\sum^\infty_{i=1}p^2(1-p)^{2(i-1)}=\frac{p^2}{1-(1-p)^2}=\frac{p}{2-p}$ I don't see how $\sum^\infty_{i=1}p^2(1-p)^{2(i-1)}=\frac{p^2}{1-(1-p)^2}$ I tried showing ...
0
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2answers
70 views

Sum of the series $\frac{1}{2.4.6}+\frac{2}{3.5.7}+\frac{3}{4.6.8}+…+\frac{n}{(n+1).(n+3).(n+5)}$.

Sum to n terms and also to infinity of the following series: $$\frac{1}{2.4.6}+\frac{2}{3.5.7}+\frac{3}{4.6.8}+.....+\frac{n}{(n+1).(n+3).(n+5)}$$ the solution provided by the book is $$S_n=\frac{...
0
votes
1answer
23 views

Induction based on combinations and binomial theorem: ${}^nC_0+{}^{n+1}C_1+{}^{n+2}C_2+\dots+{}^{n+p}C_p={}^{n+p+1}C_p$

I was looking at some questions in a Cambridge text and I reached this question however I am at it for 1 hr and can't seem to get the proof right. Any help ?
0
votes
0answers
15 views

Explicit Values of the Jacobi Theta Function [duplicate]

The sum $\sum_{n=-\infty}^{\infty}\exp(-\pi n^2) $, or $\vartheta(0;i)$ (Jacobi Theta Function) has a closed form solution of $\frac{\pi^{\frac{1}{4}}}{\Gamma(\frac{3}{4})}$ but nowhere have I been ...
-2
votes
2answers
97 views

Inequality $\sqrt{xy+yz+zx} \ge \frac {8}{15} (x+y+z)$

By Titu's inequality: $\sum_{cyc} \frac {1}{x+y} \ge \frac {(1+1+1)^2}{2(x+y+z)} = \frac {9}{2(x+y+z)}$ Then, to prove: $ \frac {3}{x+y+z} + \sum_{cyc} \frac {1}{x+y} \ge \frac {4}{\sqrt{xy+yz+zx}}$...
2
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0answers
42 views

Sum of Distinct Positive Integers Different from Sum of Any Combination of Same Integers

Find distinct $W, X, Y, Z \in\mathbb{Z}^+ :W+X+Y+Z\notin \{\sum_{a_i\in A}a_i \forall a \in A \}$ where: $A=\{$Combinations With Repetition$(W,X,Y,Z)$ of size $N\forall N\in\mathbb{Z}^+$$\}\...
0
votes
2answers
31 views

Summation of Double Exponential Series [on hold]

Is there any known closed form or tight bound analysis (big-O or big-$\Theta$) for $\sum_{i = 0}^{n} 2^{2^i}$?
1
vote
2answers
79 views

A relation involving summation and integration

Let, $\displaystyle C(x)=\sum_{n\le x}c_n$, where $\{c_n\}$ is a sequence of complex numbers ; and let $f(t)$ be a continuously differentiable function such that $\displaystyle \lim_{Y\to \infty}C(Y)f(...
2
votes
1answer
66 views

Closed form of a logarithmic sum involving binomial coefficient

I am trying to simplify an equation and in a term I have the following, $$\sum _{k=1}^n\binom{n}{k}k \log _2(k) $$ I was wondering if I can write above in closed form?
0
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0answers
65 views

Closed form of $\sum_{n = 1}^N \frac{x^{n-1}}{n!}$ [duplicate]

Is it possible to find a closed form for the following "finite sum"? $$ \sum_{n = 1}^N \frac{x^{n-1}}{n!}. $$ What I came up to: $$ \sum_{n = 1}^N \frac{x^{n-1}}{n!} = \frac{1}{x}\left( \sum_{n = 1}^...
1
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2answers
28 views

Can you give me a hint finding a sum of this arctangent?

Well, I am trying to solve find a sum of arctan, but can't find a way. Can somebody give me a hint? $\Sigma^\infty_{n=1}tan^{-1}(n+1)-tan^{-1}(n)$ I have tried to integrate it but it seemed way ...
0
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1answer
30 views

Exact formula or approximation for this sum (general harmonic series $H_{n,3}$)

I encountered the following problem in my studies. I want to calculate the requirement to the parameter $a$ for a local minimum in the function: $F(N;a) = -a*(N-1) + \sum_i^{N-1}\sum_j^i \frac{1}{j^3}...
0
votes
3answers
23 views

How to compute the summation of f(x) when the limits are in terms of y?

This is the relevant part of the question: $\sum_{y=0}^3 (x^2)$ I know the answer is $4x^2$ but I'm not certain why that is. I'm assuming that it is because there are four elements (i.e. 0,1,2 &...
6
votes
3answers
126 views

Evaluate $\sum_{n=0}^\infty (-1)^n \ln\frac{2+2n}{1+2n}$

I'm trying to evaluate: $$\sum_{n=0}^\infty (-1)^n \ln\frac{2+2n}{1+2n}$$ I'm not too sure where to start. I've tried writing it as a telescoping sum, but that doesn't work. I'm thinking it's ...
0
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0answers
25 views

Expansion of finite and infinite summations raised to a power

I need to simplify the expression written below to get the $ x $ term in its simplest form: $$ \ \left(\sum_{k=1}^b \sum_{n=0}^\infty C_n(a,k) x^\frac{n+k}{2} \right)^t ,\ $$ where $$ \ C_n(a,k)=\...
12
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1answer
294 views
+200

Mathematical Explanation of Mathematica Summation ${\sum_{n=1}^{\infty}\frac{(2n-1)!}{(2n+2)!}\zeta(2n)}$

From a mathematical point of view, what phenomena that most likely Mathematica Wolfram encountered when calculating: $$ \sum_{n=1}^{\infty}\frac{(2n-1)!}{(2n+2)!}\zeta(2n)\,=\,\color{red}{\frac{2\...
3
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1answer
23 views

Prove/disprove the following equality (sum of inverse difference products)

$$\sum_{x\in S}\frac{1}{\prod_{y\in S, y\neq x}(x-y)} = 0$$ Here, S is a finite subset of real numbers.
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2answers
28 views

How do you prove or disprove a 1:k correspondance between two series that converge to the same thing?

Suppose I have two completely different series representations of a function that can't be conventionally manipulated into each other, but converge to the same function none-the-less, like a ...
1
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2answers
55 views

Can a telescopic series with $a_n$ not tending to zero converge?

Let $\{a_n\}$ be a sequence with $a_n\to l\ne0$ as $n\to\infty$. Then, does the series $$\sum^\infty_{n=1}(a_n-a_{n+1})$$ necessarily converge? I don’t know which of the following arguments is ...
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0answers
35 views

$\sum_i ib+r,\;\; r\leq\Big[\frac{ib+r}{b}\Big]\leq ar,\; a,b, r\;\text{constant}$ [closed]

$$\sum_i ib+r,\;\; r\leq\Big[\frac{ib+r}{b}\Big]\leq ar,\; a,b, r\;\text{constant}$$ In here I wanted to find the summation boundaries for $i$ given the inequality. How to deal with inequalities with ...
4
votes
6answers
271 views

Can we prove that $\lim_{n\to\infty} \frac{1^{n_0}+2^{n_0}+\cdots+n^{n_0}}{n^{n_0+1}}$ is finite for any $n_0\in\Bbb N$ without a direct computation?

Can we prove without direct calculation that this limit is finite for any natural number $n_0 \in \mathbb{N}$? $$ \lim_{n \to \infty} \frac{1^{n_0}+2^{n_0}+\cdots+n^{n_0}}{n^{n_0+1}} $$
0
votes
4answers
70 views

Prove that $a_n \gt b_n$ $\forall $ $n \ge 6$

Given that $$a_n =\left(1^2+2^2+3^2+\cdots+n^2\right)^n$$ and $$b_n=n^n \times (n!)^2$$ Then prove that $a_n \gt b_n$ $\forall $ $n \ge 6$ My attempt: I tried using induction, but I could not ...
1
vote
2answers
20 views

Identity of the Poisson distribution

I have recently begun to study John Kingsman's "Poisson processes", and in the first chapter, the author defines $\mathbb{P}${$X$ = $n$} = $\pi_{n}$($\mu$) = $\mu^{n}e^{-\mu}/n!$ "Differentiating ...
-1
votes
1answer
56 views

Evaluating Sum of $\dfrac{i}{(-x)^i}$ [duplicate]

I would like to ask if the expression below can be simplified using standard summation properties? Or should I dive into much deeper concepts like the power series? Thank you. $$\sum_{i=1}^{n-1} \...
3
votes
2answers
47 views

Sum of Geometric Series Formula [duplicate]

I just need the formula for the sum of geometric series when each element in the series has the value $1/2^{j+1}$, where $j = 0, 1, 2, \ldots, n$. Please help. Someone told me it is: $$S = 2 - \frac{...
0
votes
2answers
21 views

Product of Summation for a single table of values.

I've been looking around but can't get a exactly clear answer on my question. I'm provided a table of values of $x_i$ and $y_i$ for $i = 1$ to $i = 5$. I'm then asked to evaluate $$\sum_{i=1}^5\sum_{...
1
vote
3answers
47 views

Simplify a sum of n products

Given the following sum formula: $\sum\limits_{i=1}^{n} (i\cdot 2^{n-i}) $ Can you help me out to a simplify the formula and provide an formula without a Sigma sign? I know that I cannot just split ...
-2
votes
1answer
72 views

Inequality about Schur

Let a,b,c be the sides of triangle such that $a+b+c=1$. Prove that $$5(ab+bc+ca)\geq18abc+a+b+c$$ I tried to prove: $$5(ab+bc+ca)\geq18abc+a+b+c$$$$10(ab+ac+bc)\geq36abc+2(a+b+c)$$$$a(5b+5c-2-12bc)+b(...
4
votes
2answers
55 views

Does changing the order of sigma notation matter?

Can you change the order of summation like this and play around? If no, then what does it change? $$ \displaystyle\sum\limits_{i=a}^{b} \sum\limits_{j=p}^{q} f(i) g(j) =\sum\limits_{j=p}^{q} \sum\...
3
votes
1answer
72 views

Fast Evaluation of Multiple Binomial Coefficients

Suppose we have a sequence of binomial coefficients. $$ S = \left\langle \binom{5}{2}, \binom{5}{3}, \binom{6}{3}, \binom{17}{14}, \binom{19}{15} \right\rangle $$ How can we efficiently evaluate all ...
2
votes
0answers
69 views

Combinatorial proof of $ \sum_{k=0}^{n}\frac{1}{\binom{n}{k}} = \frac{n+1}{2^{n+1}}\sum_{k=0}^{n}\frac{2^{k+1}}{k+1}$

A recurrence relation of $$S_n =\sum_{k=0}^{n} \frac{1}{\binom{n}{k}}$$ is $$ \frac{n+2}{\binom{n}{k}} - \frac{2n+2}{\binom{n+1}{k}} = \frac{n-k}{\binom{n}{k+1}} - \frac{n+1-k}{\binom{n}{k}}, \quad 0 ...
3
votes
1answer
82 views

Is this some magical summation?

I have been playing around with some probability calculations and somehow came to this expression $$\sum_{u_1 = 1}^n \sum_{u_2 = 1}^n \ldots \sum_{u_n = 1}^n \frac{1/u_1}{\sum_{j=1}^n 1/u_j} \prod_{i=...
-2
votes
1answer
48 views

Write $F_{1}-F_{2}+F_{3}-F_{4}+…+F_{2n-1}-F_{2n}$ as a summation

Let $F_{i}$ be the $i^{th}$ Fibonacci number (a) Write $F_{1}-F_{2}+F_{3}-F_{4}+...+F_{2n-1}-F_{2n}$ as a summation (b) Prove $F_{1}-F_{2}+F_{3}-F_{4}+...+F_{2n-1}-F_{2n}=1-F_{2n-1}$ I'm quite ...
1
vote
2answers
29 views

Need help finding a sum

I found this problem that I'm not sure how to solve. I would appreciate if anyone could point me to the right direction. I need to find the following sum: $\sum_{i+j+k=7} (-1)^i(-1)^j\frac{7!}{i!j!k!...