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

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1answer
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Summation in 104 Number Theory problems

There's a paragraph of 104 Number Theory problems, on page $9$ that says: From the formula $\prod_{i=1}^\infty\frac{p_i}{p_i-1} = \infty ,$ using the inequality $1+t \le e^t$, $t \in \mathbb{R}$ we ...
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1answer
75 views

Why does a sum of a series belong to Calculus?

When I use Microsoft Mathematics I see the sum of a series belonging to Calculus inside of the calculator pad. Can someone explain, why does the sum of a series belong to Calculus?
2
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1answer
43 views

Interesting Base summation contest math problem

The problem is as follows: Let $N_b=1_b+2_b+\cdots+100_b$ where $b$ is an integer greater than $2$. Compute the number of values of $b$ for which the sum of the squares of the digits of $N_b$ is at ...
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1answer
35 views

How prove this $\sum_{i}r_{i}x^2_{i}+2\sum_{i<j}r_{i}x_{i}x_{j}=\sum_{i}(r_{i}-r_{i-1})\left( \sum_{j=i}^{n}x_{j}\right)^2$

let $x_{1},x_{2},\cdots,x_{n}$ be real numbers,and $0=r_{0}\le r_{1}\le r_{2}\le\cdots\le r_{n}$. show ...
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1answer
63 views

Computing the sum $\sum_{n=2}^{2011}\sqrt{1+\frac{1}{n^2}+\frac{1}{(n-1)^2}}$

Find the value of $$\sqrt{1 + \frac{1}{1^2} + \frac{1}{2^2}} + \sqrt{1 + \frac{1}{2^2} + \frac{1}{3^2}} + \sqrt{1 + \frac{1}{3^2} + \frac{1}{4^2}} +...+ \sqrt{1 + \frac{1}{2010^2} + ...
5
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1answer
75 views

Find the sum : $\displaystyle\sum_{i=0}^n \frac{2^i}{1+x^{2^{i}}}$

$\displaystyle\sum_{i=0}^n \frac{2^i}{1+x^{2^{i}}}$ What technique is applicable here? I can't find a way to manipulate this sum to make it telescope. Just guide me.
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1answer
58 views

Find the value of $\lim_{n\to \infty}\sum_{k=0}^n\frac{x^{2^k}}{1-x^{2^{k+1}}}$.

If $0 \lt x \lt 1$ and $$A_n=\frac{x}{1-x^2}+\frac{x^2}{1-x^4}+.....+\frac{x^{2^n}}{1-x^{2^{n+1}}}$$ then Find $\lim\limits_{n\to \infty}A_n$.
5
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1answer
74 views

How to sum this series to infinity

How to sum the series: $$\sum _{ n=0 }^{ n=\infty }{ \frac { 1 }{ { 2 }^{ { 2 }^{ n } } } }$$ PS: Just a hint would suffice.
4
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1answer
63 views

Maximal flow and minimal cut in complete graphs

The question is as follows: We define on the complete graph $K_n$ with the vertices {$v_1, v_2, ... , v_n$} the following directions: for every j>i, the edge $v_i v_j$ is directed from $v_i$ to ...
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1answer
73 views

${n \brack k} = \sum_{m=0}^{k}2^{(k-m)^2}{n-k \brack k-m}{k \brack m}$

$${n \brack k} = \sum_{m=0}^{k}2^{(k-m)^2}{n-k \brack k-m}{k \brack m}$$ I need hint to prove this. ${n \brack k}$ is the number of $k$ dimension subspaces of $n$ dimension space over field $F_2$. I ...
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2answers
77 views

Sum of numbers on chessboard.

Consider the squares of an $8 \times 8$ chessboard filled with the numbers $1,2,3,4 \ldots ,64$ in sequential order. If we choose $8$ squares with the property that there is exactly $1$ from each ...
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1answer
31 views

Summation Induction when lower limit is not 1

The question is use induction to prove that $$\sum_{r=2}^n (r^2+r+1)r! = (n+1)^2n!-4$$ I don't understand how to even get the P1 statement since when I substitute r = 2 into the LHS and n = 1 into ...
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2answers
36 views

Proving this binomial identity

I'm required to prove the following binomial identity: $$\sum\limits_{k=0}^l {n \choose k} {m \choose l-k} = {n+m \choose l}$$ I tried various arrangements but reached nowhere. Finally I turned to ...
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1answer
21 views

Using M-Test to show you can differentiate term by term.

I have the series $\sum_{n=1}^\infty \frac{\lambda^{n-1}n}{n!}=\sum_{n=1}^\infty \frac{d}{d\lambda}\big(\frac{\lambda^n}{n!} \big)$ and I would like it to be $\frac{d}{d\lambda}\big(\sum_{n=1}^\infty ...
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3answers
135 views

How find this value $\prod_{k=1}^{\infty}\left(1+\dfrac{1}{k^5}\right)$

Find the value $$\prod_{k=1}^{\infty}\left(1+\dfrac{1}{k^5}\right)$$ I know this :How find this $\prod_{n=2}^{\infty}\left(1-\frac{1}{n^6}\right)$ and maybe can find the $2k+1$? can you someone konw ...
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2answers
42 views

Have trouble with this proof

Don´t know how to start, or apply the theorem. Aprecciate your help.
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1answer
35 views

Evaluation using Sum of Squares?

So I have a problem that looks like this: Evaluate $\displaystyle\sum\limits_{r=1}^{25} 4r^2-2r+2$ using the Sum of Squares: $\displaystyle\sum\limits_{k=1}^n \frac{n(n+1)(2n+1)}{6}$. I really have ...
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1answer
28 views

Notation: need help to understand the notation in the following formula

E is just a function such that E1=1, E2=2 and so on. But my question is the part of on "arg" and "min". 1) so "arg" stands for the angle of complex number? this doesn't make any sense. 2)If I ...
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2answers
32 views

Sum to infinity - sector [closed]

How would I start this question?
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2answers
69 views

Calculate limit of a sum

I'm currently repeating for exam and i'm stuck with limits of following two sums. $$\lim_{n\rightarrow +\infty} \sum_{k=0}^n \frac{(k-1)^7}{n^8}$$ and $$\lim_{n\rightarrow +\infty} \sum_{k=0}^n ...
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5answers
110 views

Proof via induction $1\cdot3 + 2\cdot4 + 3\cdot5 + \cdots + n(n+2) = \frac{n(n+1)(2n+7)}{6}$

(b) Prove that for every integer $n \ge 1$, $$1\cdot3 + 2\cdot4 + 3\cdot5 + \cdots + n(n+2) = \frac{n(n+1)(2n+7)}{6}$$ This is the second part of a two part question. Part (a) was the following: ...
2
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1answer
65 views

Help with $\sum_{d\mid n}τ(d)^2=\sum_{d \mid n}τ(d)^3$

I am doing some exercises on number theory on multiplicative number theoretic functions and I have some problems with the multiplication on sums like the sum $\sum_{d\mid n}(τ(d))^2$ where $d$ is a ...
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0answers
21 views

Formulating an Equation for the sum of some functions

I am trying to calculate the sum of percentages of some groups: the network is composed of groups of networks, each group is composed of nodes, I am calculating the percentage of few nodes in the ...
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0answers
48 views

Sum of all the positive integers with different digits till $100$

What is the sum of all the positive integers with 2 different digits till $100$ (without numbers with one digit and $100$) ? This was a problem I thought after hearing about Gauss and the Sum of ...
0
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1answer
17 views

Show that the normal equations are identical to $\frac{\partial}{\partial\theta_j}QS(\theta)=0~\forall~j=1,\ldots,k$

Let the quadratic sum be given by $QS(\theta)=\sum_{i=1}^{n}(y_i-x^i\theta)^2$, with $y=(y_1,\ldots,y_n)^T, \theta=(\theta_1,\ldots,\theta_k)^T$ and $$ X=\begin{pmatrix}x_{11} & \ldots ...
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0answers
29 views

Simple representation of a Sum

I have a probably pretty simple question. if I have $\sum_{i=0}^{n} 4^i$ and I want a closed representation, Wolfram Alpha gives me: $4^{n+1}/3-1/3$ Why is that? How do I get to that? Thanks!
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0answers
25 views

Sigma notation using Modulo

I've come across a step in a proof in a book on number theory that doesn't make sense to me: $$\sum_{n(mod\,p)}\frac{n(n-1)(n+1)}{p}$$ $$=\sum_{n(mod\,p)}\frac{(n+1)(n)(n+2)}{p}$$ As I understand ...
3
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1answer
57 views

Evaluate the sum $1! + 2! + 3! + \cdots + n! \le ?? < (2n)!$

How to evaluate the sum below as close as possible? $$1! + 2! + 3! + \cdots + n! \le ??? $$ Is the next evaluation $ 1! + 2! + 3! + \cdots + n! \le n n! < (2n)! $ correct?
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1answer
26 views

General Summation Formula for {0,1,2,3 … x … 3,2,1,0}

I have a series of n elements of example 1 to 10. I need a summation formula to represent this: When i = 1, add 0 When i = 2, add 1 When i = 3, add 2 When i = 4, add 3 When i = 5, add 4 When i = ...
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6answers
71 views

Why does $ 1+2+3+\cdots+p = {(1⁄2)}\cdots(p+1) $ [duplicate]

I saw this from Project Euler, problem #1: If we now also note that $ 1+2+3+\cdots+p = {(1/2)} \cdot p\cdot(p+1) $ What is the intuitive explanation for this? How would I go about deriving the ...
2
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1answer
80 views

Integrality Conjectures

Here are some interesting conjectures I would like to prove. For all positive integers $a=bc,m,n$ the following expressions are integers: $$c\sum_{k=1}^{am}k\left\{\frac{kbn}{am}\right\}$$ ...
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2answers
140 views

Dedekind Sum Integrality Result

Can we prove the following is always an integer? $$6b\sum_{k=1}^bk\left\{\frac{ka}{b}\right\}$$ where $\{x\}=x-\lfloor x\rfloor$ denotes the fractional part operator. UPDATE: Through the ...
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0answers
287 views

How find this sum $\sum_{n=1}^{\infty}\dfrac{L_{n}(2)}{n^4}=?$

Question: Find the sum $$I=\sum_{n=1}^{\infty}\dfrac{L_{n}(2)}{n^4}=?$$ where $$L_{n}(k)=1-\dfrac{1}{2^k}+\dfrac{1}{3^k}-\cdots+\dfrac{(-1)^{n-1}}{n^k}$$ since ...
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3answers
79 views

Formula for $\sum \limits_ {i=1}^n a^i $?

Does the following expression have a closed form? $$ \sum \limits_ {i=1}^n a^i $$
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5answers
131 views

How to calculate $\sum_{k=1}^n \left(k \sum_{i=0}^{k-1} {n \choose i}\right)$

How do I calculate the following summation? $$\sum_{k=1}^n \left[k \sum_{i=0}^{k-1} {n \choose i}\right]$$
7
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2answers
187 views

Evaluation of finite sum

How can one prove the following equality (for fixed positive integer $a$): $$12\sum_{k=1}^{an^2-1}k\left\{\frac{k(an-1)}{an^2}\right\}=3a^2n^4-a^2n^2-2$$ where $\{x\}=x-\lfloor x\rfloor$ denotes the ...
6
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1answer
84 views

How prove $\left(\sum\cos{\frac{2k-1}{p}\pi}\right)\cdot\left(\sum\cos{\frac{2k-1}{p}\pi}\right)$

Question:let $p$ be an odd prime number,let $A$ be the set of the (postive and less than $p$) quadratic residues modulo $p$,and $B$ be the set of the (positive and less than $p$ quadraric non-residues ...
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2answers
84 views

Use the recursive definition of summation together with mathematical induction to prove a sequence

Use the recursive definition of summation together with mathematical induction to prove that for all positive integers $n$ if $a_1, a_2,\ldots, a_n$ are real numbers, then $$\sum_{k=1}^n(3a_k - 2k + ...
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3answers
274 views

Summation of Dedekind sums to zero

I'm trying to show that: $$\sum_{a=0}^b\text{GCD}(a,b)s(a,b)=0$$ More generally, can we also show: $$\sum_{a=0}^b\text{GCD}(a,b)^ls(a,b)=0$$ where $s$ is the Dedekind sum. Any ideas?
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0answers
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A partial sum involving Euler's function

This is Exercise 2.1.17 of the book "H. Montgomery and R. Vaughan. Multiplicative Number Theory— I. Classical Theory". For $x\ge 2$, $\sum_{n\le x}\frac{\mu(n)^2}{\varphi(n)}=\log ...
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2answers
63 views

Help me with this sum: $\sum_{i=0}^\infty \frac{n}{\log^2\left(\frac{n}{5^i}\right)}$

I have to prove something but I'm stuck, I ended up with this sum. Is there any transformation I can do in the following sum? $$\sum_{i=0}^\infty \frac{n}{\log^2\left(\frac{n}{5^i}\right)}$$
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7answers
2k views

I have the pattern: 1 + 2 + 3 + 4 + 5 + 6, but I need the formula for it

I'm writing some software that takes a group of users and compares each user with every other user in the group. I need to display the amount of comparisons needed for a countdown type feature. For ...
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2answers
26 views

Problem finding explicit function without multiple $\sum$s

For $1≤n \in \mathbb{N}$, I want a function $f$ to have the following value: The sum of all possible products $a_1 * a_2 * ... * a_n$, with $a_i \in \{1, 2, 3, 4\}$ and $a_1 ≤ a_2 ≤ ... ≤ a_n$. For ...
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1answer
58 views

summation of infinite series

Find$$S=1+\frac{1}{6}+\frac{1}{18}+\frac{7}{324}+\cdots\infty$$ I could figure this pattern for denominators: $t_k=3t_{k-1}t_{k-2}$ but not sure how to proceed.
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2answers
41 views

How to find if this series is convergent or divergent

I have to find if this series is convergent or divergent. This is the series: $\sum_{n=1}^\infty{\frac{sin(5n)}{5^n}}$ I can't use the Ratio Test, and I don't know what to do with the sine in the ...
3
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0answers
65 views

Finite Trigonometric Sum

I have a dynamical system model whose equilibria depend on the solution of the following finite sum: \begin{align} \sum_{j\neq ...
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1answer
18 views

Sum with natural log

The sequence $u_n$ is defined by $u_n=0.5^n$ Find the exact value of $\sum_{0}^{10} ln(u_r)$ Is there a common ratio for this? $\frac{ln(0.5^1)}{ln(0.5^0)}$ dividing by 0 not possible..?
2
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2answers
68 views

How can I define $e^x$ as the value of infinite series?

I understand the definition of $e^x$ by limit. But I do not know how to come up with: $$e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}$$ without using Taylor series. more explicitly without using calculus. ...
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0answers
34 views

How find the sum $2\sum_{k=1}^{\infty}\sum_{i=0}^{2k-1}\frac{\binom{2k}{i}\cdot B_{i}\cdot(m-1)^{2k-i}}{2k(2k-1)m^{2k-1}}$

Find the sum $$2\sum_{k=1}^{\infty}\sum_{i=0}^{2k-1}\dfrac{\binom{2k}{i}\cdot B_{i}\cdot(m-1)^{2k-i}}{2k(2k-1)m^{2k-1}}$$ where $B_{i}$ is Bernoulli numbers. my idea: since ...
1
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1answer
29 views

Are these two summations equivalent?

Is $\sum_{y=2}^{\infty} (\frac{1}{y})(1-p)^{y-1}$ equivalent to $\sum_{y=1}^{\infty} (\frac{1}{y})(1-p)^{y}$ ?