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

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0
votes
3answers
43 views

Evaluating Nested Summations

I'm trying to evaluate the following nested summation as a function of $n$: $$\sum_{i=1}^{n-1} \sum_{j=i+1}^n \sum_{k=1}^j 1$$ So far I have: $$\sum_{i=1}^{n-1}\sum_{j=i+1}^n i+1$$ ...
1
vote
3answers
36 views

Simplify triangular sum of triangular numbers: $\sum_{i=1}^{n}(\frac12i(i+1))$

I'd like to simplify this expression, which sums up the first $n$ triangular numbers: $$\sum_{i=1}^{n}(\frac12i(i+1))$$ which is equal to: $$\sum_{i=0}^{n}((n-i)(i+1))$$ Is it even possible without ...
2
votes
2answers
52 views

Finding roots of an equation wich involves floor function

I'm trying to solve this equation $$ \left \lfloor{x +\frac{1}{100}}\right \rfloor + \left \lfloor{x +\frac{2}{100}}\right \rfloor + ... + \left \lfloor{x +\frac{223}{100}}\right \rfloor = 521 $$ I ...
3
votes
0answers
16 views

Sum of Complex Numbers and Modulus Inequality

Let $z_{1}, \dots, z_{n} \in \mathbb{C}$. Then, there exists a subset $S \subset \{1,\dots,n\}$ such that: $ \left| \displaystyle\sum_{j \in S}z_{j} \right| \geq ...
6
votes
0answers
53 views

A closed form for the following Series

I was computing some calculations, when I got stuck about a possible closed form for this series: $$S = \sum_{k = 2}^{N}\ \frac{k!}{k^k - k!}$$ I proved by hands that it's absolutely convergent by ...
23
votes
4answers
832 views

How close can $\sum_{k=1}^n \sqrt{k}$ be to an integer?

How close can $S(n) = \sum_{k=1}^n \sqrt{k}$ be to an integer? Is there some $f(n)$ such that, if $I(x)$ is the closest integer to $x$, then $|S(n)-I(S(n))|\ge f(n)$ (such as $1/n^2$, $e^{-n}$, ...). ...
6
votes
1answer
155 views
+50

Formula for cos(k*x)

I need to prove that: \begin{align} c_k =&\; \cos(k\!\cdot\!x)\\ c_k :=&\; c_{k-1} +d_{k-1}\\ d_k :=&\; 2d_0\!\cdot\!c_k +d_{k−1}\\ d_0 :=&\; −2\!\cdot\!\sin^2{(x/2)}\\ \end{align} ...
3
votes
3answers
76 views

Deriving the formula for the $n^{th}$ tetrahedral number

$T(n)$ is $n^{th}$ triangular number, where $T\left(n\right)=\frac{n^2+n}{2}$ And from other sources I know the $n^{th}$ tetrahedral number is $G\left(x\right)=\sum ...
1
vote
2answers
66 views

Prove that $\frac 12\leq \frac{1}{n+1}+\frac {1}{n+2}+\frac{1}{n+3}…+\frac{1}{n+n}$

How do I prove $$\frac 12\leq \frac{1}{n+1}+\frac {1}{n+2}+\frac{1}{n+3}...+\frac{1}{n+n}$$ without using induction? Note that clearly $n\neq 0$ Thanks for any help!!
17
votes
5answers
1k views

Evaluate $\sum\limits_{k=1}^n k^2$ and $\sum\limits_{k=1}^n k(k+1)$ combinatorially

$$\text{Evaluate } \sum_{k=1}^n k^2 \text{ and } \sum_{k=1}^{n}k(k+1) \text{ combinatorially.}$$ For the first one, I was able to express $k^2$ in terms of the binomial coefficients by ...
6
votes
5answers
116 views

Prove by mathematical induction: $\frac{1}{n}+\frac{1}{n+1}+\dots+\frac{1}{n^2}>1$

Could anybody help me by checking this solution and maybe giving me a cleaner one. Prove by mathematical induction: $$\frac{1}{n}+\frac{1}{n+1}+\dots+\frac{1}{n^2}>1; n\geq2$$. So after I check ...
9
votes
4answers
222 views

Proving that $\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}>\frac{13}{24}$ by induction. Where am I going wrong?

I have to prove that $$\frac{1}{n}+\frac{1}{n+1}+\dots+\frac{1}{2n}>\frac{13}{24}$$ for every positive integer $n$. After I check the special cases $n=1,2$, I have to prove that the given ...
-1
votes
2answers
107 views

What $+1+1+\cdots$ really equals

$1+1+1+\cdots$ is clearly a divergent series, so you'd say that it tends towards infinity? Through analytic continuation of the zetafunction the value $-1/2$ could be assigned the sum, right? But if ...
1
vote
1answer
38 views

Closed-form Solution of Log Sum

I have the series: $$\sum_{i=1}^{i=10^N} \log_5 i$$ I'm trying to figure out how to get the closed-form solution to this problem. I entered it into WolframAlpha and got that it equals: $ ...
0
votes
5answers
66 views

Closed-form Solution to a Sum

I have some math questions for a programming course where it says to provide closed-form solutions for a list of sums. I've never taken an algorithms course, so I'm not quite sure what I'm doing. I ...
2
votes
2answers
48 views

Factorial Summation Definition

A while back I found the series $$\sum_{k=0}^n \binom n k (-1)^k (x+k)^n = (-1)^n n!$$ while messing around in Algebra class (specifically when $n$ is any natural number and $x$ is any real number) I ...
-1
votes
0answers
43 views

Closed form for $\left(\sum_{k=0}^n\frac{x^k}{k!}\right)^p$

The expression for the p-th power of the sum of the first $n+1$ powers of x is given analytically by ...
5
votes
1answer
89 views

How to solve this hard sum problem?

$$\sum _{ x=1 }^{ \infty }{ \frac { 3{ x }^{ 2 }+12x+16 }{ { \left( x\left( x+1 \right) \left( x+2 \right) \left( x+3 \right) \left( x+4 \right) \right) }^{ 3 } } } =\frac { 1 }{ 4{ (a!) }^{ b } } ...
1
vote
1answer
18 views

Function Equivalent to the Maximum Operator?

All numbers are real, WLOG positive. $A + B + ... + N = T$ and $A' + B' + ... + N' = T$ I'm trying to figure out some function, f, such that if $f(A,B,... ,N) > f(A',B',...,N')$ then, ...
0
votes
1answer
38 views

Show that $\sum _ {i=1} ^{\lg n - 1} \frac 1 {\lg n - i} = \sum _{i=1} ^{\lg n - 1} \frac 1 i$ [closed]

I couldn't understand this summation: $$\sum\limits_{i=1}^{\lg n - 1} \frac{1}{\lg n -i} = \sum\limits_{i=1}^{\lg n - 1} \frac{1}{i} .$$ How did author transform LHS to RHS? Can you describe in ...
5
votes
3answers
38 views

Combinatorial argument for $\sum\limits_{k=i}^{n}\binom{n}{k}\binom{k}{i} = \binom{n}{i}2^{n-i}$

I need to show that $$\sum\limits_{k=i}^{n}\binom{n}{k}\binom{k}{i} = \binom{n}{i}2^{n-i}$$ I know that $\displaystyle \binom{n}{k}\binom{k}{i}$ is counting the number of ways to pick $k$ elements ...
14
votes
3answers
2k views

What do $\{ceps_q\}_{q=0}^Q$ and $\{a_q\}_{q=1}^p$ mean?

As a programmer who hasn't had any higher mathematical training, I sometimes find mathematical equations described in books or online that I'd like to implement in my programs, but they have symbols ...
1
vote
0answers
26 views

Possible closed form or approximation?

Does it have some closed form or approximation ? I tried on my own but i am not getting any idea regarding this. $$\sum_{k_1=k}^{M}\sum_{k_2=k}^{M}\frac{k_1^{-\gamma} k_2^{-\gamma} ...
2
votes
1answer
65 views

Summation from Right to Left changes Accuracy?

I was looking at The accuracy from left to right and that from right to left of the floating point arithmetic sums which was asking for the accuracy of $$\sum_{k=1}^n\frac{1}{k^2}$$ from right to ...
0
votes
1answer
21 views

how can $((\frac1N \sum z_i )- I )^2 = \frac1{N^2}(\sum (z_i - I)^2 )$?

$$\left(\left(\frac1N \sum z_i \right)- I \right)^2 = \frac1{N^2}\left(\sum (z_i - I)^2 \right)$$ How does this work? Or is there an error? I thought that you could not pull the sum out of the ...
2
votes
1answer
50 views

Does $\sum_{i=1}^n \frac{1}{i^2}=O(\ln(n))$?

I was looking at Why $\sum_{i=1}^n \frac{1}{i} =\mathcal O(\ln(n))$?. And there it was proved that $$\sum_{i=1}^n \frac{1}{i} =\mathcal O(\ln(n))$$ My question is that does this also stand for ...
0
votes
1answer
34 views

Differentiating $- \sum_{n \in \mathbb{Z}^2} e^{i n \cdot \alpha}\int_0^E\frac{1}{4\pi t}\exp({\omega^2 t - \frac{|x - n - y|^2}{4t^2}})dt$ wrt $x$?

I have a formula for the Ewald method which can be used to speed up computations when working with periodic Green's functions. I will need to take the derivative of the function $G(x, y)$ with respect ...
32
votes
25answers
27k views

Proof for formula for sum of sequence $1+2+3+\ldots+n$?

Apparently $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$. How? What's the proof? Or maybe it is self apparent just looking at the above? PS: This problem is know as "The sum of the first $n$ positive ...
5
votes
1answer
99 views

Random Wolfram|Alpha identity related to $\sum_{k = 1}^{\infty}{\tan^{-1}}{\frac{1}{k^2}}$

I was watching a Numberphile video (on how $\tan^{-1}{1} + \tan^{-1}\frac{1}{2} + \tan^{-1}\frac{1}{3} = \frac{\pi}{2}$) and I thought about whether the series $$\sum_{k = ...
0
votes
1answer
29 views

Calculate the sum under gcd

I have a question: How can I calculate $$\sum_{i=1\atop \gcd(i,75)=1}^{75} i=?$$
-2
votes
1answer
45 views

Dealing with phi function property

If $n=2^kN$, where $N$ is odd, then $$\sum_{d\mid n}(-1)^{n/d}\phi(d)=\sum_{d\mid 2^{k-1}N}\phi(d)-\sum_{d\mid N}\phi(2^kd)$$ I have no idea how to seperate things inside the left side. In a ...
4
votes
2answers
183 views

Euler-Maclaurin Summation

Using EM summation formula estimate $$ \sum_{k=1}^n \sqrt k $$ up to the term involving $\frac{1}{\sqrt n}$ My attempt is $$ \sum_{k=1}^n \sqrt k = \frac{2 \sqrt{n^3}}{3} -\frac{2}{3} + \frac 1 ...
204
votes
8answers
12k views

The length of toilet roll

Fun with Math time. My mom gave me a roll of toilet paper to put it in the bathroom, and looking at it I immediately wondered about this: is it possible, through very simple math, to calculate (with ...
0
votes
1answer
14 views

Difficulty simplifying nested sums with different variables

I'm trying to work out an algorithm analysis problem, and I'm having some difficulty determining how a jump is made between two steps in the answer. $$ \begin{align} ...
0
votes
1answer
41 views

Find the closed form for the double sum $ \sum_{1\leq j \leq k \leq n }3^k=\sum_{j=1}^n \sum_{j=k}^n 3^k$

Find the closed form for the double sum $$ \sum_{1\leq j \leq k \leq n }3^k$$ Here is my attempt: $$ \sum_{j=1}^n \sum_{j=k}^n 3^k $$ What should I do next to get the closed form? Please help me
2
votes
1answer
25 views

Easy way of seeing if swapping summation is ok? (Generating functional derivation of Bell numbers)

On page 21 of his book generatingfunctionology (available for free on the author's homepage), the author rearranges the summations in the following way: ...
2
votes
2answers
113 views

Why does $\frac{d}{d\theta} \left(\theta\prod\limits_{i=1}^nx_i\right) = \sum\limits_{i=1}^n\ln x_i$

Is this just the product rule? I have this in my notes but I didn't think anything of it and now I'm wondering how this happens? Edit: Im working with maximum likelihood estimation and in my notes I ...
9
votes
0answers
89 views

Find value to the summation : $\sum_{n =1}^\infty \dfrac 1 {5^{n+1}-5^n+1}$

$$\sum_{n = 1}^\infty \dfrac 1 {5^{n+1}-5^n+1}$$ I can factorize denominator to $4\times5^n+1$ to confirm the series does not diverge, But how do I calculate its actual sum? The series is not a ...
0
votes
2answers
18 views

Infinite convergent sum with central binomial coefficient over k

Given the following sum: $$0.5\cdot\sum\limits_{k=0}^\infty \frac{1}{k+1}\binom{2k}{k}\cdot(0.25)^{k}$$ I know that the sum is supposed to converge to $1$. How would I go about evaluating it to get ...
0
votes
2answers
55 views

How to find total number of sum of consecutive number of $n$? [duplicate]

How many ways are there to write $n$ as the sum of consecutive positive integers? Example: $15$ has $3$ consecutive sums: $1+2+3+4+5=15$ $7+8=15$ $4+5+6=15$
16
votes
1answer
396 views

How find this $\sum\limits_{n=0}^{\infty}(-1)^{n}\left(\frac{1}{n+1}-\frac{1}{n+2}+\frac{1}{n+3}-\cdots\right)^{2}$

Find the value $$\sum\limits_{n=0}^{\infty}(-1)^{n}\left(\frac{1}{n+1}-\frac{1}{n+2}+\frac{1}{n+3}-\cdots\right)^{2}$$ This problem is from ...
1
vote
1answer
30 views

How to neatly summarize indexes of a matrix where there are a lot of i's x j's [closed]

As you can see from the subject line, I can't even think of the word of what I need to do. I am trying to write in text that I multiplied columns of a matrix (n columns, i = 1:n). There are many ...
0
votes
1answer
34 views

Simultaneous equation with summation and square - how to solve?

$\mathbf{p}$ is a vector with dimension: $x \times 1$ $\mathbf{d}$ is a vector with dimension: $1 \times y$ $\mathbf{V}$ is a matrix with dimension: $x \times y$ $y \geq x$ $\mathbf{d}$ and ...
1
vote
1answer
26 views

Sum of Legendre symbols: $\sum _{n=1}^p \left(\frac{an+b}{p}\right)=0$

If $(a,p)=1$ and $p$ is an odd prime, prove the Legendre symbol sum $$\sum _{n=1}^ p \left(\frac{an+b}{p}\right)=0.$$ Where $b$ is any integer. I know the fact that $\sum_{a=1}^p \left( \frac{a}{p} ...
2
votes
4answers
265 views

Can we assign a value to the sum of the reciprocals of the natural numbers?

I know the sum of the reciprocals of the natural numbers diverges to infinity, but I want to know what value can be assigned to it. ...
5
votes
1answer
57 views

Prove inequality $1 < \frac{1}{n} + \frac{1}{n+1} + \ldots + \frac{1}{3n-1} < 2$

Prove the inequality $1 < \frac{1}{n} + \frac{1}{n+1} + \ldots + \frac{1}{3n-1} < 2$ For all $n \in \mathbb{N}$ I've done the right hand side, but can't do the left side of the inequality. For ...
2
votes
0answers
20 views

summation combinatoric again with floor function

$\sum_{n=1}^{33}\binom{3n}{\left \lfloor 1.5n-0.5 \right \rfloor}= ...$ $\binom{3}{\left \lfloor 1 \right \rfloor}+\binom{6}{\left \lfloor 2.5 \right \rfloor}+\binom{9}{\left \lfloor 4 \right ...
3
votes
3answers
95 views

Proof that $\sum_{i=0}^n {n\choose i}2^{i}=3^{n}$ [duplicate]

The following sum came up in a combinatorial argument. I know what it equals thanks to Wolfram Alpha, but I'm not sure how to show it $$\sum_{i=0}^n {n\choose i}2^{i}=3^{n}$$
3
votes
1answer
28 views

Proving that maximizing a sum of functions of different independent variables is equivalent to maximizing each function

Let $$ \pi = f_1(x_1) + f_2(x_2) + f_3(x_3) + \dots + f_n(x_n) = \sum_{i=1}^n f_n(x_i) $$ where $f_i$ denote different functions and $x_i$ denote different independent variables Would proving that ...