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
36 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 ...
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} ...
1
vote
1answer
24 views
9
votes
0answers
94 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
1answer
43 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: ...
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: $1+2+3+4+5=15$ $7+8=15$ $4+5+6=15$
-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 ...
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 ...
0
votes
1answer
45 views

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

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 ...
0
votes
1answer
35 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 ...
3
votes
1answer
40 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 ...
0
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0answers
11 views

Showing summation of disjoint sets can be split

Take three sets $X,Y,Z \subseteq V$ $$\sum_{u\in X\cup Y}\sum_{v\in Z}f(u,v)=\sum_{u\in X}\sum_{v\in Z}f(u,v) + \sum_{u\in Y}\sum_{v\in Z}f(u,v) \text{ if $X \cap Y=\varnothing$}$$ It seems ...
2
votes
0answers
29 views

Factorial ratio sum of finite series

Given: $ S = \sum_{i=1}^{n-1}{i! \over n!} $ How would I find the sum for an arbitrarily large $n$ ? Example: $n=5$ $ S = \frac{1!}{5!} + \frac{2!}{5!} + \frac{3!}{5!} + \frac{4!}{5!} = 0.275 $
0
votes
1answer
29 views

Non-infinite geometric sum; does not start at 0 or 1

It's bee a long time since I've worked with sums and series, so even simple examples like this one are giving me trouble: $\sum_{i=4}^N \left(5\right)^i$ Can I get some guidance on series like this? ...
1
vote
1answer
31 views

Solving $\max_{x\in\prod_{i=1}^n s_i} \sum_{i=1}^n f(x_i)$ by maximizing for each $i$ individually.

First, I will clarify some of the notation: $$ x_i \in S_i,\; i\in \{1,2,\dots, n\} \quad x\in S, \quad S\equiv \prod_{i=1}^nS_i \text{ (direct product set)} $$ So basically, we have $x \in S$ which ...
1
vote
2answers
46 views

Find the generalized sum of $1+2(2)+3(2)^2+4(2^3)+…+n(2^{n-1})$ [duplicate]

Find the generalized sum of $1+2(2)+3(2)^2+4(2^3)+...+n(2^{n-1})$ I rewrote the above sequence into: $\sum_{k=1}^{n} k(2^{k-1})$. The sequence looks like a hybrid of the summation $\sum_{k=1}^{n} ...
210
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
2answers
47 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
2answers
41 views

Limit of power series with L'Hospital

Calculate the given limit: $$\lim_{x\to 0} \frac{1}{1-\cos(x^2)}\sum_{n=4}^\infty\ n^5x^n$$ First, I used Taylor Expansion (near $x=0$): $$1-\cos(x^2)\approx 0.5x^4$$ I'm now quite stuck with the ...
0
votes
0answers
31 views

Number of ways a dice can roll every side equally many times for the first time after x rolls

This problem is best viewed as a walk on a $d$-dimensional integer lattice with integer steps corresponding to various results of a dice roll. For a normal 6-sided dice, these would be ...
1
vote
3answers
52 views

Find $\lim\limits_{n\to\infty}\left(\frac{1}{\sqrt{n}}\sum\limits_{k=1}^{n}\frac{1}{\sqrt{2k}+\sqrt{2k+2}}\right)$

I don't know how to find the sum of $\sum\limits_{k=1}^{n}\frac{1}{\sqrt{2k}+\sqrt{2k+2}}$. After rationalization we have ...
0
votes
1answer
49 views

Binominal expression simplification

I need to simplify the expression $$\sum_{k = 1}^{10} k\binom{10}{k}\binom{20}{10 - k}$$ Thank you.
8
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3answers
148 views

Find the sum $\sum _{ k=1 }^{ 100 }{ \frac { k\cdot k! }{ { 100 }^{ k } } } \binom{100}{k}$

Find the sum $$\sum _{ k=1 }^{ 100 }{ \frac { k\cdot k! }{ { 100 }^{ k } } } \binom{100}{k}$$ When I asked my teacher how can I solve this question he responded it is very hard, you can't solve it. I ...
3
votes
2answers
63 views

How to simplify $\lim_{n\to \infty}\sum_{r=1}^n \tan^{-1} \dfrac{2r+1}{r^4+2r^3+r^2+1}$

$$\lim_{n\to \infty}\sum_{r=1}^n \tan^{-1} \dfrac{2r+1}{r^4+2r^3+r^2+1}$$ How am I supposed to do it? One thing I see here is $$\lim_{n\to \infty}\sum_{r=1}^n \tan^{-1} \dfrac{2r+1}{(r^2+r)^2+1}$$ ...
4
votes
3answers
68 views

Find the sum of the $\sum_{m=k}^{+\infty}\binom{m}{k}(1-p)^k\cdot p^{m-k}$

Let $0<p<1$,Find the sum $$\sum_{m=k}^{+\infty}\binom{m}{k}(1-p)^k\cdot p^{m-k}$$
1
vote
1answer
31 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 ...
3
votes
4answers
95 views

Converting Σ i(i + 1) into a formula, given this hint

The given summation is: $$\sum_{i=1}^n i(i+1)$$ The goal is to convert it into a formula which only uses n. Solving this, I got the answer: $$\frac{n}{3}(n+1)(n+2)$$ However, I don't believe the ...
0
votes
0answers
32 views

Strange use of sigma notation in computability

Ok everyone, so I was reading about computability when I came across the following- ''Suppose that $f(x, z)$ is any function; the bounded sum $\sum_{z<y} f(x, z)$ is a function of $x, y$ given by ...
-1
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1answer
20 views

What are the steps in between to derive at the solution of this summation? [closed]

What are the steps in between to derive at the solution of this summation?
1
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4answers
32 views

Summation of more than one series

How can I find $1+3x+6x^2+10x^3+.... $ for $x=6/7$. I have seen that this can be written as a sum of three series $\sum_0^\infty [(n^2/2)x^n+(3/2)nx^n+x^n]$. But I do not know how to proceed further.
0
votes
2answers
17 views

Prove minimum of $\sum_{i=1}^n=S_i$ where all $S_i$ are limited by $x \le S_i \le y $

Sorry if this has already been asked or answered somewhere on the net. I have a set of values $S=\{S_1,S_2,S_3,... S_n\} $ where $x \le S_i \le y $. S has an unknown number of discrete members, ...
3
votes
1answer
31 views

Summation of $\sum_{k=1}^{n}\left \lfloor \log _{m}k \right \rfloor$ and $\sum_{k=1}^{n}\left \lceil log_{m}k\right \rceil$

$$\sum_{k=1}^{n}\left \lfloor \log _{m}k \right \rfloor$$ $$\sum_{k=1}^{n}\left \lceil log_{m}k\right \rceil$$ I found myself stuck trying to solve these two summations but i can't make any progress. ...
0
votes
2answers
53 views

Sum of $\sum_{i=0}^n k^i$ when $k=1$

I have this sequence as a function of $n$: $$\sum_{i=0}^n k^i$$ which is equal to $$\frac{k^{n+1}-1}{k-1}$$ for each $k\in\Bbb{Z}\backslash\{1\}$. For $k=1$, the above formula returns $\frac00$. Is ...
0
votes
1answer
21 views

When is a multiple sum given in closed form?

Let $d$ be a positive integer and $a>0$. Consider a following multiple sum: \begin{equation} {\mathcal S}^{(d)}_a(j) := \sum\limits_{0 \le j_1 \le j_2 \le \dots \le j_d \le j} \prod\limits_{l=1}^d ...
1
vote
1answer
21 views

How can I change the bounderies of the following sum

i want to change the lower bound of the summation with $k=0$. $$\sum_{k=2}^{r+s-1} { \{ \begin{pmatrix} k-1\\r-1\\ \end{pmatrix} +\begin{pmatrix} k-1\\s-1\\ \end{pmatrix} \} } H(k,r+s-k) $$ In here, ...
0
votes
3answers
50 views

How to find sum of equation from 1 to N

I understand that the sum of n from $1$ to $n$ is $\frac{n(n+1)}{2}$. I'm trying to figure out the sum from $1$ to $n$ of the following expression $\frac{L(n-1)}{R}$ where $L$ and $R$ are unknown ...
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votes
0answers
58 views

Given $2a_{n+1}={a_n}^2+1$, $a_1=3$, find the sum $\Sigma_{n=1}^\infty \frac{1}{1+a_n}$

Given $2a_{n+1}={a_n}^2+1$, $a_1=3$, find the sum $\Sigma_{n=1}^\infty \frac{1}{1+a_n}$. It is pretty hopeless to find the general formula for $a_n$. Rearrange gives ...
1
vote
1answer
30 views

Prove an equality involving Fejer Kernel

Define $$V_n(\theta) = \sum_{k=-n-1}^{n+1} e^{in\theta} + \sum_{k=n+2}^{2n+1}\frac{2n+2-k}{n+1}(e^{ik\theta}+e^{-ik\theta})$$ I want to show that $$V_n(\theta) = 2K_{2n+1}(\theta) - K_n(\theta)$$ ...
0
votes
1answer
23 views

Find $m$ such that $\sum_{k=1}^{m-1}\frac{\sin k}{k}-\csc(\frac{1}{2})\frac{3}{m}\gt 0$

Use a calculator to find an integer $m$ such that $$\sum\limits_{k=1}^{m-1}\frac{\sin k}{k}-\csc\left(\frac{1}{2}\right)\frac{3}{m}\gt0$$ Is there a calculator online that I can plug this into?
0
votes
1answer
33 views

generalised log sum inequality

The log sum inequality states that $ \sum_i a_i\ln\frac{a_i}{b_i}\geq a\ln{\frac{a}{b}}$ where $a=\sum_i a_i$ and $b=\sum_i b$. Is there a generalisation (with whatever conditions) that extends it ...
1
vote
1answer
48 views

Product of two sums, one finite and one infinite

I'm working on a problem and I'm not sure how to find the product of these two sums: $\left(\sum_{k=0}^{\infty}\text{something}\right)\left(\sum_{k=n}^{n}\text{something else}\right)$ The ...
0
votes
1answer
19 views

Equating sums, and removing the summation sign

In a problem I'm working on, I develop an expression: $$\sum_{i=1}^NB_i =K_4 \frac{\sum_{i=1}^NM_i}{K_2+\sum_{i=1}^NM_i}$$ What I really want is an expression for an individual $B_i$. Through various ...
0
votes
0answers
27 views

Changing summation order. When is $\sum_{j=1}^{n}\ \sum_{i=1}^{\infty} a_{ij} = \sum_{i=1}^{\infty}\ \sum_{j=1}^{n} a_{ij} $?

Let's say I have an expression like that: $\sum_{i=1}^{\infty}\ \sum_{j=1}^{n} a_{ij} $ where $a_{ij}$ includes $n$, $i$ and $j$. In what circumstances can I safely switch the sigmas around, i.e. make ...
0
votes
1answer
12 views

Fundamental Period of a Summation

How would I find the fundamental period of a summation such as this one? Any hints/suggestions?
1
vote
2answers
38 views

Bound on sum of combinations

I came across the following inequality $\sum_{i=0}^D \binom N i \le N^D+1$. I am not sure how to prove this. I tried to do it by induction on $D$, and started with observing the values of sum for ...
3
votes
2answers
45 views

Finding combinatorial sum

How to compute $$\sum_{k=0}^{n} \left( k^2 \cdot \binom{n}{k} \cdot 3^{2k}\right)? $$ I have no idea other than guessing the answer and proving it by induction.
2
votes
4answers
266 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. ...