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

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18
votes
1answer
130 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
votes
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 $k+1$ ...
-1
votes
2answers
46 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 } ...
0
votes
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
vote
1answer
53 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
vote
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
36 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
193 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 ...
0
votes
0answers
41 views

Why does the harmonic series diverge? [duplicate]

I had a little doubt over a small matter: if $(1/n)\to0$ even as $n\to \infty$ then how come the summation $$\sum_{n=1}^{\infty} \frac 1 n$$ is divergent (as stated in my textbook).
0
votes
0answers
14 views

Simplifying a sum and expressing its propotionality

I have a simple question about this expression: $e^{\frac{\tau}{2}\sum_{i=1}^{n} (y_i-\mu)^2-\frac{\mu^2}{2}}$. How can this: $e^{\frac{\tau}{2}\sum_{i=1}^{n} (y_i-\mu)^2-\frac{\mu^2}{2}} \propto ...
2
votes
4answers
117 views

Find the sum of binomial coefficients

Calculate the value of the sum $$ \sum_{i = 1}^{100} i\binom{100}{i} = 1\binom{100}{1} + 2\binom{100}{2} + ...
4
votes
1answer
39 views

Splitting an infinite unordered sum (both directions)

This question is a follow-up to this. Let $A, B_1, B_2, \dots$ be countable sets such that $\bigcup_{i \in \mathbb{N}}B_i = A$ and the $B_i$'s are disjoint. Let $f : A \to \mathbb{R}$ take elements ...
8
votes
0answers
214 views

How to simplify this combinatorial expression?

Find \begin{eqnarray} \sum_{j\in\mathbb{N}}(n-2j)^k\binom{n}{2j-m} \end{eqnarray} Note that this question is a generalization of this one. I tried to imitate the steps in the answer given in that ...
5
votes
4answers
166 views

Proving that $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots+\frac{1}{2n-1}-\frac{1}{2n}=\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2n}$

I have written the left side of the equation as $$\left(1+\frac{1}{3}+\frac{1}{5}+\cdots+\frac{1}{2n-1}\right)-\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\cdots+\frac{1}{2n}\right).$$ I don't know how ...
1
vote
1answer
50 views

What is the sum of integers between 1 and 200 inclusive that are divisible by both 4 and 5?

What is the sum of integers between 1 and 200 inclusive that are divisible by both 4 and 5? Now I attempt the question of taking the integers to be divisible by 20(LCM of 4 and 5) and get the number ...
2
votes
1answer
52 views

Find $a$, given $y(n)=x(n)+ax(n-d)$, interesting question

Me and two friends of mine are working on a project (scholarly purposes only). The goal of this project is to clean an audio signal (speech, a song, anything audio) of echo. Generally speaking, if ...
1
vote
0answers
38 views

Can this combinatoric sum be simplified?

Base cases: $F(n,k,d) = 0$ if $d=0$ and $n>0$ $F(n,k,d) = 1$ if $n=0$ Expression: $$F(n,k,d) = \sum_{s=0}^{\min(k,n)}\binom{n}{s}F(n-s,k,d-1)$$ I am trying to compute the value of $F(n,k,10)$ ...
3
votes
0answers
112 views

Richert’s theorem breaks down for $ n = 11 $

In 1949 H.-E.Richert proved (1) that every positive integer typeset structure is a sum of distinct primes. For more information please look at (2), and (3). However, if you consider $ n = 11 $, you ...
1
vote
1answer
67 views

Can the sum of 3 unique primes be expressed as the sum of 2 primes?

Let's consider the example, $$ 3 + 11 + 19 = 33 \\ 2 + 31 = 33 $$ we can see that there are cases where the sum of 3 primes be expressed as the sum of 2 primes. However, I couldn't find a case ...
2
votes
0answers
73 views

Find the general formula for summation of square root of rational function

We were given a problem statement saying: Using discrete sum, find the area of the unbounded region limited by the curve $y^2=\frac{x(x-3)^2}{6-x} , x\ge3$ and its asymptote. I've made the following ...
13
votes
9answers
3k views

Has anyone heard of this maths formula and where can I find the proof to check my proof is correct? $\sum^n_{i = 1}i + \sum^{n-1}_{i=1}i = n^2$

The formula basically is: The sum of all integers before and including $n$, plus all the integers up to and including $n-1$. This will find $n^2$. $$ \sum^n_{i = 1}i + \sum^{n-1}_{i=1}i = n^2 $$
3
votes
2answers
31 views

Maximizing Theta in a Summation Formula

I need to take the first derivative of $$\sum Y_i (\log(\Theta )) +(n-\sum Y_i)(\log(1-\Theta )) $$ with respect to theta, and then solve for theta. I believe this is my derivative... $$\frac ...
4
votes
2answers
49 views

Is there any way to compute these sums quickly?

I have a sum of the following form (all numbers are positive integers): $$F(p) = \sum_{x=1}^{N} a_x x^p $$ Where $N$ and all $a_x$ terms are known/fixed constants. However I need to be able to ...
1
vote
2answers
57 views

Evaluate $\binom{n}{1}\alpha_1+\binom{n}{2}\alpha_2+\binom{n}{3}\alpha_3+…+\binom{n}{n}\alpha_n$

If $\alpha_1,\alpha_2,.....,\alpha_n$ are the n;$n^{th}$ roots of unity then$\binom{n}{1}\alpha_1+\binom{n}{2}\alpha_2+\binom{n}{3}\alpha_3+......+\binom{n}{n}\alpha_n$ ...
0
votes
1answer
39 views

Simulate sum of N dice throws

For starters let me apologise if it isn't proper forum or if it was asked, I'm not very good at probability and what more I'm not familiar with proper english terms, so I could miss something. I'm ...
0
votes
1answer
30 views

Summation Formula

I'm trying to do build some stats, but I don't know formulas very well. What would be the appropriate formula for this scenario: $1$ Person is worth $\$1$/minute. Starting at $5$ People. ...
2
votes
2answers
45 views

Proving $\sum_{j=1}^n \frac{1}{\sqrt{j}} > \sqrt{n}$ with induction

Problem: Prove with induction that \begin{align*} \sum_{j=1}^n \frac{1}{\sqrt{j}} > \sqrt{n} \end{align*} for every natural number $n \geq 2$. Attempt at proof: Basic step: For $n = 2$ we have $1 ...
1
vote
1answer
47 views

Resources on Variants of the Clausen Functions

I am interested in locating more information about the Clausen functions. Specifically I am looking for the closed forms of the Gl-type (or Sl-type as they are sometimes called) and the alternating ...
0
votes
3answers
91 views

Calculate the sum of this series

$$ \sum_{n=1}^\infty \frac{1}{n^2 3^n} $$ I tried to use the regular way to calculate the sum of a power series $(x=1/3)$ to solve it but in the end I get to an integral I can't calculate. Thanks
5
votes
3answers
63 views

Finding the general formula for $a_{n+1}=2^n a_n +4$, where $a_1=1$.

Problem: Find the general formula for $a_{n+1}=2^n a_n +4$, where $a_1=1$. Find the sum of its first $2n$ terms with odd subscript. My effort: It seems to me that $a_{n+1} / ...
3
votes
2answers
83 views

Sum of Squares in terms of Sum of Integers

We know that the sum of squares can be expressed as a multiple of the sum of integers as follows: $$\begin{align} \sum_{r=1}^n r^2 &=\frac 16 n(n+1)(2n+1)\\ &=\frac {2n+1}3\cdot \frac ...
6
votes
1answer
154 views

Find the sum of the series below

Find the sum $$(1\cdot2)+(1\cdot3)+(1\cdot4)+\cdots+(1\cdot2015)+(2\cdot3)+(2\cdot4)+\cdots+(2\cdot2015)+\cdots+(2014\cdot2015)$$ What I have tried... We are looking for ...
0
votes
1answer
29 views

Help evaluating a partial sum with factorials and binomial coefficients

I come from a CS background and had to contend with a problem similar to this one. Essentially, I want a general-case estimate on how many rolls I'd have to make to land on the same number twice with ...
3
votes
4answers
74 views

Prove by induction: $\frac{1}{2!}+\frac{2}{3!}+\cdots+\frac{n}{(n+1)!}=\frac{n!-1}{n!}$

Prove $$\frac{1}{2!}+\frac{2}{3!}+\cdots+\frac{n}{(n+1)!}=\frac{n!-1}{n!}.$$ My problem with this is that it doesn't hold for the base case: $n=1$. This question is from the book "Abstract ...
0
votes
1answer
23 views

Summation of finite power seires

Is it possible to find a close form solution for $S_1$. $S_1$ is defined as follows: $S_1=\sum_{k=b}^{\infty}\frac{x^k}{k!}$ ; Where $0<x<b<\infty$ If $b=0$ then $S_2 = e^x$. But how do we ...
1
vote
1answer
21 views

Ensemble average of square of fluctuations proof

The ensemble average of a random variable $x$ is denoted as $X$ or $\left \langle x \right \rangle$, and is defined as: $$ X = \left \langle x \right \rangle = \lim_{N \to \infty} \frac{1}{N} ...
-3
votes
3answers
62 views

Find the sum of all products of two distinct naturals, neither exceeding 2015. [closed]

Find the sum $$(1\cdot2)+(1\cdot3)+(1\cdot4)+\cdots+(1\cdot2015)+(2\cdot3)+(2\cdot4)+\cdots+(2\cdot2015)+\cdots+(2014\cdot2015)$$ any help? I tried with telescope but got nothing
0
votes
0answers
50 views

Derivative of a summation series

Let $f_{n}(x)=x+(1-x)x^2+(1-x)(1-x^2)x^3+\cdots+(1-x)(1-x^2)\cdots(1-x^{n-1})x^n;\quad n\geq4$ then $f'(x)=\text{ ?}$ $$(A)\qquad (1-f_{n}(x)) \left(\sum\limits_{r=1}^n ...
6
votes
4answers
57 views

Proving $\sum_{i=1}^n 2^i = 2^{n+1} - 2$ using strong induction [duplicate]

I just started learning proof by induction in class, but got a problem requiring proof by strong induction. Here is the problem. Prove by strong induction: $$\sum_{i=1}^n 2^i = 2^{n+1} - 2$$ ...
-1
votes
1answer
53 views

How would I go about creating the summation formula

Consider the following program segment, where i, j , k, n, and counter are integer variables and the value of n (a positive integer) is set prior to this segment. ...
0
votes
1answer
33 views

How to get initial digits from a sum before adding?

How would you add digits in such a way that when having a result sum, the initial digits that were added could be extracted/calculated? For example, when having the following random digits 132748107 ...
2
votes
1answer
39 views

Closed form for a floor sum

I want to compute $$\sum_{i=a}^b \left\lfloor \frac{i}{k} \right\rfloor$$ Where $k < b < \infty$, and $a > 0$. I don't know where to begin (or if there's a closed form, for that matter), so ...