Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.

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0
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1answer
27 views

Weighted sum of angles modulo $\pi/2$

Angle modulo $\pi /2$ means: $(a+ \pi /2) \mathbin{\%} \pi/2=a$, $a \in [0, \pi/2)$, which could be illustrated as a ‘modulo circle’ in the following figure. How to calculate the weighted sum of a ...
1
vote
2answers
78 views

Prove the Identity $\pi=2- \sum_{1}^{∞} \frac{(-1)^m}{m^2-\frac{1}{4}} $

By considering the fact that $f(\pi/2)=1$, prove the identity $\pi=2- \sum_{1}^{∞} \frac{(-1)^m}{m^2-\frac{1}{4}} $ This question was is a subsection in a chapter on Fourier series, can I use my ...
3
votes
2answers
53 views

Sum over two binomials identity

So while trying to count the number of configurations in a statistical mechanics research problem I come across this lovely sum: $$\sum_{i=0}^k \binom{i+r}{r} \binom{k-i+r}{r}$$ I scoured the ...
0
votes
1answer
18 views

Change of indices in a double summation

By using a "smart" change of indices $i$ and $j$, I'm trying to show that \begin{equation} \sum_{i=1}^{N}\sum_{j=1}^{N}q_{i}q_{j}a_{i}\left(f_{i}f_{j}^{'}-f_{i}^{'}f_{j}\right) = ...
2
votes
2answers
40 views

Binomial sum with two parameters

Let $m$ and $n$ be two integers. Evaluate $$S_{m,n}=\sum_{j=0}^{m} (-1)^j \binom{m}{j}\binom{mn-jn}{m+1}$$ At first, for $n=2$ I got $S_{m,2}=2^{m-1}m$, for $n=3$ I obtained $S_{m,3}=3^m m$, then I ...
13
votes
3answers
228 views

What is $\sum_{r=0}^n \frac{(-1)^r}{\binom{n}{r}}$?

Find a closed form expression for $$\sum_{r=0}^n \dfrac{(-1)^r}{\dbinom{n}{r}}$$ where $n$ is an even positive integer. I tried using binomial identities, but since the binomial ...
2
votes
1answer
26 views

Prove or disprove: $ \sum_{b \vee d = x} \tau(b) \tau(d) = \tau(x)^3$

Can somebody prove or disprove? Let $\tau$ be the divisors function, so that $\tau(6) = \#\{ 1,2,3,6\} = 4$ $$ \sum_{b \vee d = x} \tau(b) \tau(d) = \tau(x)^3$$ Here I am using $b \vee d = ...
0
votes
2answers
56 views

How I can simplify this double sum?

How I can simplify this double sum: $$S=\sum_{k=1}^{n-1}\sum_{l=1}^{k}\frac{4^{2kl}×5^{k^{2}}-4^{k^{2}}×5^{2kl}}{4^{l^{2}}×5^{l^{2}}×4^{k^{2}}×5^{k^{2}}}$$
0
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2answers
42 views

How this is true $(1-\sum_{i \geq 3}2^{-i}) = (3/4)$?

I'm reading a paper A comparison of two lower bound methods for communication complexity, P.43 $$(1-2^{-1}) \times (1-2^{-2}) \times (1-\sum_{i \geq 3}2^{-i}) = (1/2)(3/4)(1-\sum_{i \geq 3}2^{-i}) = ...
2
votes
2answers
57 views

Why sigma notation?

Repeated union is written as: $$\bigcup_{i=0}^na_i$$ Repeated logical conjunction is: $$\bigwedge_{i=0}^na_i$$ Etc. So why isn't repeated addition: $$\operatorname{\huge+}\limits_{i=0}^n{}^{\Large ...
1
vote
1answer
51 views

Sigma Notations

I have troubles understanding the sigma notation. If for example we have $c_i$ as $$c_i=\frac {x_i-x}{\sum(x_i-x)^2}$$ $$\sum c_i=\sum\frac{x_i-x}{\sum(x_i-x)^2}$$ Do we distribute the sigma to both ...
0
votes
1answer
59 views

How to calculate this sum?

Let $x_1,\cdots,x_k$ be numbers between 0 and 1. Then is it possible to get explicit expression for the following sum:$$\sum_{n_1,\cdots,n_k\geq 1} x_1^{n_1}\times C_{n_1+n_2}^{n_2}\times ...
11
votes
2answers
489 views

Limit of an Integral, Then taking Sum

I am given that $I_n=\int^1_0x^ne^x\,dx$ Now, how can I find the value of the following limit: $$\lim_{n\to\infty}\left(\sum_{k=1}^{n}\frac{I_{k+1}}{k}\right)$$ I suppose solving for $I_n$ is that ...
1
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2answers
45 views

Binomial coefficient as a summation series proof?

Alright, so I was wondering if the following is a well known identity or if its existence provides any real benefits other than serving as a time-saver when dealing with higher values for ...
5
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4answers
125 views

Surprising Summation (2): $\frac14 \sum_{i=1}^{2n}i(2n-i+1)=\sum_{i=1}^n i^2$

$$\underbrace{\frac14 \sum_{i=1}^{2n}i(2n-i+1)}_{A} =\underbrace{\sum_{i=1}^n i^2}_{B} =\underbrace{\frac 16 n(n+2)(2n+1)}_{C}$$ Transform $A$ directly into $B$ without expanding to $C$ but using ...
1
vote
2answers
47 views

inverting the summation

Let $\{a_n\}$ be a sequence of non-negative real numbers. Then, how can one prove rigoroulsy that $$ \sum_{n=1}^\infty \frac{1}{n^2} \sum_{j=1}^n a_j = \sum_{j=1}^\infty a_j \sum_{n=j}^\infty ...
0
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0answers
30 views

Equation with a summation (Use of Harmonic series)

I have a sum $\sum\limits_{i=j}^k \dfrac{1}{i^s}$ and a constant $j$. I would like to determine $k$ such that $\sum\limits_{i=j}^k \dfrac{1}{i^s}=C$ where $C$ is a constant $< 2$. How can I ...
0
votes
1answer
30 views

Super algebraic decaying series

For $a>0$, $b > 1$ and $c \geq 1$ it holds $$F(a,b,c):=\sum_{j=a+1}^\infty \exp(-b \log(j)^c) \leq \int_{a}^\infty \exp(-b \log(x)^c) \, d x < \infty.$$ I am looking for upper bounds on $F$. ...
2
votes
1answer
38 views

Find this sum $\sum_{k=1}^{2015}k\lfloor \log_{2}{k}\rfloor\equiv \pmod {1000}$ [closed]

Find $$\sum_{k=1}^{2015}k\lfloor \log_{2}{k}\rfloor\equiv \pmod {1000}$$ consider $2^{i-1}\le k\le 2^i-1$,then $\lfloor\log_{2}{k}\rfloor=i-1$,This problem have simple methods?
-1
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2answers
134 views

Finding the sum of this series: $1+\frac 12 + \frac 13 + \cdots + \frac 1{50}$ [duplicate]

I need to find the sum of this series: $1+\frac 12 + \frac 13 + \cdots + \frac 1{50}$ Please help me find the sum of this series.
1
vote
1answer
22 views

Changing indices of a sum (Basic question)

I was working through some proofs and it came to a point where I had to change the index of a sum which started as sigma (from k=0 to n-1) Substituting in i=k+1 So that means, my k=0 would become ...
2
votes
1answer
66 views

Upper bounding a tricky sum

For a problem in probability, I'm trying to find an upper bound for $$ \sum_{d=0}^k\binom{k}{d}\gamma^d(1-\gamma)^{k-d}\left(1-p^d(1-p)^{k-d}\right)^m$$ which will help me analyze what values of ...
7
votes
2answers
163 views

Value of $\frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+\cdots+\sqrt{10+\sqrt{99}} }{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+\cdots+\sqrt{10-\sqrt{99}}}$

Here is the question: $$\frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+\cdots+\sqrt{10+\sqrt{99}} }{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+\cdots+\sqrt{10-\sqrt{99}}} = \;?$$ (original image) I ...
0
votes
0answers
35 views

Find the Fourier transform of the given memory function in the limit volume $V\rightarrow\infty$

The memory function is given by, \begin{equation} \mu (t)=(8\pi e^{2}/3V)\sum_{\vec{k}}|f_{\vec{k}}|^{2}\cos (ckt) \end{equation} where $V$ is the volume, $f_{\vec{k}}$ is the form factor. In this ...
5
votes
5answers
111 views

Surprising Summation $\sum_{r=1}^n(n-r+1)(2r-1)$

Reduce this ordinary-looking summation to a surprisingly familiar summation in the fewest possible steps, preferably without brute force expansion. (Please also read my comment below for context) ...
4
votes
2answers
195 views

Is the limit $ \lim_{n\to \infty}\left(\sum^{n}_{r=0} \binom{n}{r}\big/{n^{r}(r+3)}\right)$ rational or irrational?

How can I prove that the result of the following limit is rational/irrational?$$ \lim_{n\to \infty}\left(\sum^{n}_{r=0} \frac{\binom{n}{r}}{n^{r}(r+3)}\right)$$ Would solving this limit satisfy? How ...
2
votes
1answer
41 views

Division in Summations

Suppose $a_n=\dfrac{2^n}{n(n+2)}$ and $b_n=\dfrac{3^n}{5n+18}$. I need to find the value of: $$\displaystyle\sum_{n=1}^{\infty}\dfrac{a_n}{b_n}$$ I think this problem is meant for me to compute ...
0
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0answers
20 views

Partial sum of a product over an arbitrary sequence.

Below is an equation a friend showed me, but was unable to prove. After struggling with it for a bit I was unable to as well. After failing to show this for, say, N=2 Im pretty sure the equation is ...
5
votes
1answer
77 views

Why is this sum well-defined?

Let $$S=\mathbb{N}[1/2]$$ be the set of rational numbers greater than $0$ which has a power of $2$ as its denominator. Let $R$ be any commutative ring. Let us consider $R^S,$ the infinite direct ...
1
vote
1answer
115 views

Prove that $\sum_{k=1}^{n} \frac1{\sin^2 \frac{\left( 2k-1\right)\pi}{4n+2}}=2n\left( n+1\right)$

Prove that $$\frac{1}{\sin^{2}\frac{\pi }{4k+2}}+\frac{1}{\sin^{2}\frac{3\pi }{4k+2}}+\frac{1}{\sin^{2}\frac{5\pi }{4k+2}}+\cdots+\frac{1}{\sin^{2}\frac{(2k-1)\pi }{4k+2}}=2k(k+1)$$
4
votes
1answer
69 views

Integration of $\sin(\theta)$

I hope I'm not asking a silly question. We can integrate $\sin(\theta)$ simply by the following identity: $$\int_0^\frac{\pi}{2} \sin\theta\ \mathsf d\theta = \left[-\cos\theta \vphantom{\frac 1 ...
3
votes
2answers
78 views

Double Summation Trick

I have seen a couple of times the trick where $\displaystyle\sum_{i=1}^\infty \sum_{j=i}^\infty f(j)$ becomes $\displaystyle\sum_{j=1}^\infty \sum_{i=1}^j f(j)$ How does this work? I am so confused. ...
0
votes
0answers
8 views

How to calculate and more importantly visualise Prefix Sum for an 3 , 4 and n-dimensional array?

I was trying to solve this problem . http://codeforces.com/problemset/problem/372/B I tried reading on http://wcipeg.com/wiki/Prefix_sum_array_and_difference_array but I still ...
0
votes
0answers
10 views

how to simply this summation expression using moments generation function?

The P(x) represents the Moments Generating Function, as $L^{*}(z) = \sum_{x=0}^{\infty}{ P(x) z^{x} }$. The expression needed to be simply is as follows: ...
18
votes
1answer
123 views

Does the sum converge for all values of $a$?

Here is the sum (for which $a$ is the variable): $$a+\sin(a)+\sin(\sin(a))+\cdots$$ Does the sum always converge for all values of $a$? So far, this is what I have done: 1) I plugged in many ...
0
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0answers
28 views

Sum over product of residues modulo two different bases

Let $M(x,y)$ be a modulo function. Specifically, $$ M(x,y) = \begin{cases} x, & \text{when } -\lfloor \frac y2\rfloor \leq x \lt \lfloor \frac y2\rfloor \\ M(x-y,y), & \text{when } x \geq ...
2
votes
0answers
78 views

Is there a formula for $1+\sqrt{2}+\sqrt{3}+\cdots+\sqrt{N}$? [duplicate]

Is there a known formula to the sum $$1+\sqrt{2}+\sqrt{3}+\cdots+\sqrt{N}$$ where $N$ is some natural number? Thanks
12
votes
4answers
2k views

Why is this sum zero?

I have been looking at the following sum (for any positive integer $n$) $$\left(1-\frac{1^2}{n}\right) + \left(1-\frac{1}{n}\right)\left(1-\frac{2^2}{n}\right) + ...
1
vote
3answers
38 views

Show that $1/\sqrt{1} + 1/\sqrt{2} + … + 1/\sqrt{n} \leq 2\sqrt{n}-1$ [duplicate]

Show that $1/\sqrt{1} + 1/\sqrt{2} + ... + 1/\sqrt{n} \leq 2\sqrt{n}-1$ for $n\geq 1$ I attempted the problem but I get stuck trying to show that if the statment is true for some $k\geq1$ then $k+1$ ...
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votes
2answers
45 views

Does limit exist for the following expression?

If limit exists, then what is its value? And if it does not exist then can we find where does this expression tends as $ n \to \infty$. The expression : $\lim\limits_{n \to \infty } ...
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2answers
50 views

Efficiently evaluate a triple nested summation [closed]

How do you efficiently evaluate the following nested sum, perhaps as a product of matrices and/or vectors: $$ ...
1
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1answer
52 views

How to interpret double summation with same subscripts

How would I interpret the following summation (where $r_{k}$ is a function that produces a scalar): $f^{int}_i = \frac{(\kappa \mathbf{b})_{i} + (\kappa \mathbf{b})_{i + 1}}{2} ...
2
votes
4answers
70 views

Simplifying $\sum_{i=1}^{n-2}i(n-1-i)$

I have been trying to simplify $\sum_{i=1}^{n-2}i(n-1-i)$ i.e - remove the summation, put it in polynomial form Since $i$ is the changing variable, I don't think this is possible. I also know that ...
1
vote
1answer
34 views

How do I find the sum of $\sum\limits_{k=1}^\infty{\frac{k}{2^{k+1}}}=1$? [duplicate]

As shown in the title, how do I find the sum of: $$\sum\limits_{k=1}^\infty{\frac{k}{2^{k+1}}}=1$$
1
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2answers
40 views

Finite sums of integers and similar problems: book request

I recently learned about Faulhaber's formula, which says that for each integer $p \geq 1,$ we can simplify the finite sum $\sum_{k \in \mathbb{N}}[k<n]k^p$ so that it becomes an (integer-valued) ...
1
vote
2answers
35 views

Adding matrix elements using single summation

Is it possible to add all elements of square matrix using single summation notation, instead of using double summation?
1
vote
3answers
55 views

How $\sum_{j=2}^{n}{1}$ is equal to `n−1` in this example?

This is the example I am talking about.
2
votes
4answers
88 views

Explanation of a method to compute $\sum_{k \le n} k^2$

I was searching for methods to compute $\sum_{k\le n} k^2$. I stumbled across this (which is an answer provided by Gareth Rees to this question). "..Represent $k^2$ in terms of falling powers ...
0
votes
1answer
192 views

How are these equations equal?

I am reading CLRS 3rd edition(Wikipedia page) on page 26, author deduced a formula for the running time of ...
1
vote
1answer
88 views

Find the upper bound of $\limsup_{M\to\infty}\left|\frac{1}{\sqrt{M}}\sum\limits_{m=1}^{M}\sqrt{m}\cos(m\theta)\right|$

I want to calculate the upper bound of this $\limsup_{M\to\infty}\left|\frac{1}{\sqrt{M}}\sum\limits_{m=1}^{M}\sqrt{m}\cos(m\theta)\right|$. There is a constraint that $\theta\neq2n\pi$ where $n$ is ...