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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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Summation involving the closest integer to $\sqrt n$

Let $f(n)$ be the integer closest to $\sqrt n$. Evaluate $$\sum_{n=1}^\infty\frac{\left(\frac32\right)^{f(n)}+\left(\frac32\right)^{-f(n)}}{\left(\frac32\right)^n}$$ In this question, I was able to ...
cende's user avatar
  • 9
1 vote
1 answer
53 views

Compute the value of a double sum

I need some help computing a(n apparently nasty) double sum: $$f(l):=\sum_{j = \frac{l}{2}+1}^{l+1}\sum_{i = \frac{l}{2}+1}^{l+1} \binom{l+1}{j}\binom{l+1}{i} (j-i)^2$$ where $l$ is even. I'm not ...
Matt M's user avatar
  • 27
-1 votes
0 answers
39 views

How to give this sum a bound?

Let $x,y\in\mathbb{Z},$ consider the sum below $$\sum_{x,y\in\mathbb{Z}\\ x\neq y}\frac{1}{|x-y|^{4}}\Bigg|\frac{1}{|x|^{4}}-\frac{1}{|y|^{4}}\Bigg|,$$ is there anything I could do to give this sum a ...
Chang's user avatar
  • 329
3 votes
3 answers
52 views

Prove that $\frac{(n + 1)!}{((n + 1) - r)!} = r \sum_{i=r - 1}^{n} \frac{i!}{(i - (r - 1))!}$

I Need Help proving That $$\frac{(n + 1)!}{(n - r + 1)!} = r \cdot \sum_{i=r - 1}^{n} \frac{i!}{(i - r + 1)!}$$ Or in terms of Combinatorics functions: $P_{r}^{n+1} = r \cdot \sum_{i = r-1}^{n} {P_{r-...
BGOPC's user avatar
  • 179
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0 answers
27 views

Abel summation on smooth piecewise function

After reading this article on non-analytic smooth functions, I wondered if is possible to apply the Abel Summation Formula to such a function. The respective Wikipedia articles assert that Abel ...
Richard Burke-Ward's user avatar
1 vote
1 answer
59 views

Sum $\sum_{n=0}^\infty x^n\sin(x^n),\quad x\in(-1,1)$

I am trying to find a closed-form expression for the sum $$\sum_{n=0}^\infty x^n\sin(x^n),\quad x\in(-1,1)$$ Can anyone help me / provide any useful insights? What I've Tried I know a couple methods ...
Da Monster's user avatar
1 vote
1 answer
27 views

Subset of index that minimizes a sum of real values

Given a series of real numbers $c_1, \ldots, c_n$ with $ n \in \mathbb{N} $, is there an algorithm or method to find the subset of indices such that the absolute value of the sum of the values within ...
mcmat23's user avatar
  • 1,070
0 votes
1 answer
62 views

Sums involving reciprocal of primes

I am interested in obtaining an upper bound for $$ \sum_{j=1}^N 1/p_j $$ and upper+ lower bound for $$ \sum_{p|K} 1/p. $$ Here $p_j$ is the j-th prime and $p$ is prime. I was able to find an ...
Johnny T.'s user avatar
  • 2,903
4 votes
1 answer
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Sum of reciprocal Bernoulli numbers

What is sum of the Bernoulli numbers? discusses the sum of the Bernoulli numbers, using divergent sum methods since the Bernoulli numbers grow exponentially. This exponential growth makes it so that ...
D.R.'s user avatar
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0 answers
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Which sums of factorials of distinct positive integers are Poulet-numbers? [closed]

A Poulet-number is a composite positive integer $N$ satisfying $2^{N-1}\equiv 1\mod N$ , in other words a weak Fermat-pseudoprime to base $2$. Which sums of the factorials of distinct positive ...
Peter's user avatar
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1 vote
0 answers
33 views

Sum with constraints in maple or mathematica [closed]

I'm looking for a code in Maple or Mathematica to evaluate and give a list of terms in expressions like $\newcommand{\on}[1]{\operatorname{#1}}$ $$ \sum_{a\ +\ b\ +\ c\ =\ 6}\on{f}\left(a\right)\on{f}\...
wkmath's user avatar
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-3 votes
0 answers
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Set of integers lesser than factorial of the digit summation [closed]

For which integers n would the factorial of the sum of its digits be less than n? I can’t seem to find a generalized rule, at least not one that’s reliable. (And that’s just for base ten.)
Nalacram's user avatar
1 vote
1 answer
146 views

Find sum of factorials divisible by the largest possible prime squared

Let $n$ be a positive integer. Consider the following maximization problem : Use each of the factorials $1,2,3!,\cdots ,n!$ at most once such that the resulting sum is divisible by $p^2$ , where $p$ ...
Peter's user avatar
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0 votes
2 answers
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Show that $S(M) \leq 5a^2$

The problem Let $ABCDA'B'C'D'$ be a cube of edge a. On $[BC']$ we consider a point $M$ and write $S(M)=AM^2+CM^2+D'M^2$ a) Show that $S(M) \leq 5a^2$ b) Determine the position of point $M$ so that $S(...
IONELA BUCIU's user avatar
2 votes
0 answers
50 views

Proving an interesting sequence [duplicate]

So a while ago I was learning about these sums, $$\sum_{i=1}^ni=\frac{n(n+1)}{2}$$ $$\sum_{i=1}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$$ $$\sum_{i=1}^ni^3=\frac{(n(n+1))^2}{4}$$ I wondered if I could find a ...
Vedant Lohan's user avatar
0 votes
1 answer
91 views

Determine all possible values ​of the sum $S=a+b+c+d$

The problem Let $a$ and $b$ be two integers for which the interval $(a,b]$ contains 20 integers and let $c,d$ be two natural numbers for which the interval $(c,d)$ contains 24 natural numbers. ...
IONELA BUCIU's user avatar
2 votes
0 answers
100 views

$x_{1}=\frac{1}{x_{1}}+x_{2}=\frac{1}{x_{2}}+x_{3}=\ldots=\frac{1}{x_{n-1}}+x_{n}=\frac{1}{x_{n}}$; prove that $x_{1}=2\cos\frac{\pi}{n+2}$ [closed]

If $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$ are $n\geq 2$ positive real numbers such that $ x_{1}=\frac{1}{x_{1}}+x_{2}=\frac{1}{x_{2}}+x_{3}=\ldots=\frac{1}{x_{n-1}}+x_{n}=\frac{1}{x_{n}}$, prove that $...
Sushil's user avatar
  • 141
-4 votes
1 answer
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What is the meaning of $E[V|Sell] = \sum_{V_i=V_{min}}^{V_i=V_{max}}V_iPr[V=V_i|Sell]$. [closed]

$E[V|Sell] = \sum_{V_i=V_{min}}^{V_i=V_{max}}V_iPr[V=V_i|Sell]$. I am new to probability, so detailed explanation would really help. Also what is $Pr[V = V_i|Sell]$ in this context?
Raj Kumari's user avatar
-2 votes
2 answers
58 views

Asymptotic behavior of $\sum_{k=1}^n k^{n-k+1}$ as $n$ goes to $+\infty$ [duplicate]

I am in front of the sum $\sum_{k=1}^n k^{n-k+1}$. I find no way to evaluate it in closed form. Is there any? Is there a way to find an equivalent as $n$ goes to infinity?
Aristodog's user avatar
  • 369
-1 votes
0 answers
26 views

Three things to show in $\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}A_nB_k$ [closed]

I would like to find a very formal way $100$% approved by professionals to show that in this sum of sums with $n$ and $k$ both integers defined both in $[0,\infty)$: $\sum_{n=0}^{\infty}\sum_{k=0}^{\...
lazare's user avatar
  • 277
4 votes
1 answer
102 views
+50

Is there a compact form of $\frac{2}{n+1}\sum_{k=1}^n\frac{\sin^2(\pi jk/(n+1))}{z-2\cos(\pi k/(n+1))}$ in terms of $z$, $n$, $j$?

I would like to evaluate the following sum: $$S_j(z,n)=\frac{2}{n+1}\sum_{k=1}^n\frac{\sin ^2\left(\frac{\pi j k}{n+1}\right)}{z-2 \cos \left(\frac{\pi k}{n+1}\right)}$$ where $z\in\mathbb{C}$ is an ...
papad's user avatar
  • 123
1 vote
2 answers
114 views

Closed form for the recurrence $S_n = 1 + S_{n-1} + \frac{2}{n} S_{n-2}$, where $S_1=1$ and $S_2=2$?

How would you go about getting an expression for $S_n$ where $S_1=1$, $S_2=2$, and $S_n = 1 + S_{n-1} + \frac{2}{n} S_{n-2}$? I'm using this to try and solve a separate problem which involves the ...
ojt's user avatar
  • 75
3 votes
0 answers
161 views

Closed form for $\sum_{k=n}^{2n-1} k(-1)^k \sum_{i=k}^{2n-1} {2n \choose i+1} {i \choose n}$?

I've found this sum: $$S(n) = \sum_{k=n}^{2n-1} k(-1)^k \sum_{i=k}^{2n-1} {2n \choose i+1} {i \choose n}$$ The inner sum is elliptical iirc, but perhaps the double sum has a nice expression. We can ...
hellofriends's user avatar
  • 1,930
8 votes
2 answers
666 views

Positive sum can always be presented as a sum with strictly positive incremental sub-sums

I am trying to prove the following (which to my knowledge is just a "conjecture", I have not seen this proved anywhere or mentioned as a theorem so it very well may be a false statement): ...
giorgio's user avatar
  • 583
1 vote
2 answers
46 views

Why the "$+1$" in $\sum_{x=\lceil u\rceil}^ne^{-f(x)}\leq\int_u^\infty e^{-f(x)} \; dx + 1$?

So I have been reading Lattimore's book on Bandit Algorithms, and while proving the lemma for Asymptotic UCB, he bounds the sum of $e^{-f(x)}$ as follows: $$\sum\limits_{x = \lceil u \rceil}^{n} e^{-f(...
tango's user avatar
  • 13
-1 votes
0 answers
96 views

Proof of $\sum_{n=1}^{∞}\frac{(-1)^{n+1}}{\sum_{k=1}^{n}k^2} = 6(\pi - 3)$ [closed]

I was solving the question Prove that $$ \sum_{n=1}^{∞}\frac{(-1)^{n+1}}{\sum_{k=1}^{n}k^2} = 6(\pi - 3)$$ How to solve this summation?
Chetan's user avatar
  • 69
2 votes
0 answers
73 views

Abel-Plana vs Euler-MacLaurin summation for $S = \sum\limits_{k=1}^{\infty}\left[ 2 \pi k - 2 - 4k \tan^{-1}(2k)\right]$

I have a sum as follows: $$S = \sum\limits_{k=1}^{\infty} 2 \pi k - 2 - 4k \tan^{-1}(2k) \approx -0.250854$$ $$S=2+\sum\limits_{k=0}^{\infty} 2 \pi k - 2 - 4k \tan^{-1}(2k)$$ Applying Euler-MacLaurin ...
Srini's user avatar
  • 862
0 votes
0 answers
21 views

Converting a double product into a single product

Consider the product $$ P=\prod_{i=1}^n\prod_{j=1}^k X_{ij}^{y_{ij}}(1-X_{ij})^{1-y_{ij}}, $$ where $y_{ij}=0\mbox{ or }1$ $\forall i$. I would like to convert this double product into a single index ...
Ludwig's user avatar
  • 179
0 votes
1 answer
76 views

How to solve this limit $\lim_{n\to \infty } \sum_{k=0}^n 1/[(2k+1)(2k+3)]$ [closed]

Help please: $$ \mbox{I tried to do it like this}\quad \lim_{n \to \infty}\sum_{k = 0}^{n}{1 \over \left(2k + 1\right)\left(2k + 3\right)}$$ but I don't know how to continue.
intenziven's user avatar
2 votes
1 answer
81 views

If $M:=\sum\limits_{k=1}^{\frac{n(n+1)}{2}}\lfloor\sqrt{2k}\rfloor$ How to Find $\frac{n^3+2n}{M}$?

$$M:=\sum\limits_{k=1}^{\frac{n(n+1)}{2}}\lfloor\sqrt{2k}\rfloor$$ Find $\frac{n^3+2n}{M}$ This problem was on a problem book. It is easy to find $M$ If $n$ is odd, $\ m=\frac{n+1}{2} $ and $$M= \...
pie's user avatar
  • 6,536
1 vote
0 answers
64 views

Prove that $ \sum _{j=0}^n \frac{(-1)^j (1-j)^n}{j! (n-j)!} = 1 $ [duplicate]

I have shown via geometric methods that the below result is actual, but I cannot prove it via algebraic means. I have tried a simple proving approach through induction but cannot get it right. I'm ...
G. Brickhill's user avatar
-4 votes
0 answers
34 views

Show that, whatever are three non-zero consecutive natural numbers, the sum of their inverses represents a mixed periodic decimal fraction. [closed]

Show that, whatever are three non-zero consecutive natural numbers, the sum of their inverses represents a mixed periodic decimal fraction. so we have to show $\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}=...
IONELA BUCIU's user avatar
-3 votes
0 answers
49 views

Summation of a $x^5$ [duplicate]

What is the summation of the series $x^5$ for $x=1$ to $x=n$? I want the actual formula for it and also the proof. I would also like to know how we can use the proof of summation of $x^5$ to find the ...
Surajsing Rajput's user avatar
2 votes
1 answer
56 views

Proof of $\frac{\Phi_n'(x)}{\Phi_n(x)} = \frac{1}{x^n-1}\sum_{k=1}^{n}c_k(n)x^{k-1}$

Let $c_k(n)$ denote Ramanujan's sum, and $\Phi_n(x)$ be the $n$th cyclotomic polynomial. Prove that $$\frac{\Phi_n'(x)}{\Phi_n(x)} = \frac{1}{x^n-1}\sum_{k=1}^{n}c_k(n)x^{k-1}$$ My attempt was to ...
Mako's user avatar
  • 690
1 vote
0 answers
42 views

Integral involving a rational function inside a hypergeometric ${}_2F_1$

I want to know if the following integral can be expressed in terms Hypergeometric (or any other rational) functions \begin{equation} \int_{0}^1 \int_0^1 dr\, dt\,r^{-1-i p} t^{-1-i p} (1-r)^{i p+2 q-1}...
QFTheorist's user avatar
3 votes
3 answers
180 views

How to find the correct constant term with Euler-Maclaurin formula, $\sum_{j=1}^n j\log j$

Question: Is there a way to find the complete asymptotic expansion (by deriving the correct $O(1)$ term) via the Euler-Maclaurin formula? Let $$s_n=\sum_{j=1}^n j\log j$$ Fix $m\ge 1$. By the Euler-...
bob's user avatar
  • 2,227
-4 votes
0 answers
43 views

Find $n^ 1 +n^2+\dots. + n^k$? [duplicate]

How do I find $n^ 1 +n^2+\dots. + n^k$? I found a post that asks about $n ^ 1 + n ^ 2 +\dots+ n^{n - 1}$, but I only want until $n^k$, and I can't apply the answer of that post to fit in my use. ...
Đỗ Quốc Khánh's user avatar
-5 votes
1 answer
49 views

Question about concrete mathematics double summation derivation [closed]

How did the author in the image convert the summation into a double summation? I can see how the double summation turns into the sum of squared integers but how would you go about converting the sum ...
adeldude13's user avatar
1 vote
0 answers
48 views

Is there a rule for using parentheses or brackets after the summation symbol to indicate what is included in the sum? [duplicate]

Using parentheses or brackets removes ambiguity but is it necessary?
Alex's user avatar
  • 19
3 votes
3 answers
383 views

$\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$ as a limit of a sum

Working on the same lines as This/This and This I got the following expression for the Dilogarithm $\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$: $$\operatorname{Li}_{2} \left(\frac{1}{e^{\...
Srini's user avatar
  • 862
0 votes
0 answers
94 views

$\operatorname{Li}_{2} \left(\frac12 \right)$ vs $\operatorname{Li}_{2} \left(-\frac12 \right)$ : some long summation expressions

Throughout this post, $\operatorname{Li}_{2}(x)$ refers to Dilogarithm. While playing with some Fourier Transforms, I came up with the following expressions: $$2 \operatorname{Li}_{2}\left(\frac12 \...
Srini's user avatar
  • 862
2 votes
2 answers
177 views

Is there a closed formula for this sum? [duplicate]

The sum is $$f_{n}=\sum_{k=1}^{n}\arctan\left(\frac{1}{\sqrt{k}}\right)$$ I figured I need a closed formula for this or for the cosine of this whole expression in order to get a polar representation ...
עמית חי לרמן's user avatar
1 vote
1 answer
67 views

Change order of summation

I need to change the summation order in the sum: $$ \sum\limits_{n=0}^{\infty} \sum\limits_{k=0}^{ n m } F(k, n - mk) $$ In Srivastava's book, I came across a similar formula \begin{array}{c} \sum\...
Leox's user avatar
  • 8,204
-1 votes
2 answers
62 views

Upper bound on the finite sum of $\sum_j x^j/j!$ [closed]

How can I derive an upper bound on the following finite summation, \begin{equation} S = \sum_{j=1}^k \frac{x^j}{j!}, \end{equation} where $0 < x$, in terms of $x$ and $k$ (it's perfectly fine to ...
mike's user avatar
  • 23
6 votes
0 answers
78 views

Showing that $\sum_{i = 1}^N \sum_{j = N - i + 1}^N \frac{1}{ij} = \sum_{i = 1}^N \frac{1}{i^2}$ without induction

It's straightforward to show via induction that $$ \sum_{i = 1}^N \sum_{j = N - i + 1}^N \frac{1}{ij} = \sum_{i = 1}^N \frac{1}{i^2}\text{.} $$ (For example, given the table $$ \begin{array}{|c|c|c|c|}...
Fred Akalin's user avatar
1 vote
3 answers
201 views

Summation of arithmetic series [closed]

How to represent a sum of $n$ items of an arithmetic series with the use of the sigma notation? For the sum of the 10 first items, is the below one correct? $$ \color{gray}{ \sum_{\{{x_i: x_{i-1}+k \}}...
Damian Czapiewski's user avatar
-2 votes
2 answers
107 views

Reference for ${}_nC^0 + {}_nC^1 + {}_nC^2 + \cdots + {}_nC^n = 2^n$ [closed]

How can I find, or what is a good reference for: $${}_nC^0 + {}_nC^1 + {}_nC^2 + \cdots + {}_nC^n = 2^n$$ I could write References [1] Binomial sums, Binomial Sums -- from Wolfram MathWorld but I need ...
Mocean's user avatar
  • 15
0 votes
0 answers
51 views

Derivative with respect to an index within a summation

First note that $$ \sum_{n=0}^{\infty} \frac{\left(\frac{1}{2}\right)_{n}}{n!} \, x^n = \frac{1}{\sqrt{1-x}}, $$ where $(a)_{n}$ is the Pochhammer notation. The background is how to evaluate the ...
Leucippus's user avatar
  • 26.5k
7 votes
1 answer
372 views

Which numbers are sums of finite numbers of reciprocal squares?

Question: Is there a “nice” characterization of rational numbers $q$ for which $q$ can be written as $$q = \frac{1}{n_1^2} + \frac{1}{n_2^2} + \dots + \frac{1}{n_k^2}$$ for distinct natural numbers $...
templatetypedef's user avatar
-1 votes
1 answer
46 views

Resources to master summation symbol [closed]

I noticed that I have some difficulties to use the summation tools( change of index, double or multiples summation...). Do you have some resources or book to master this topic. I am using concrete ...
Moi Moi's user avatar

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