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

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IMPROVED - Proving that a statistics is not sufficient (Gaussian case).

Let $X=(X_1,...,X_n)$ be i.i.d. $N(0,\sigma^2)$. How to show that $$\frac{2}{n}\sum_{i=1}^{n}X_i$$ is not a sufficient statistic? I have already proven that $\max_{i=1,...,n}X_i$ is a sufficient ...
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1answer
24 views

$S_n = \sum_{k=1}^{n} (\sqrt{1 + \frac{k}{n^2}} - 1)$ Show that $\lim_{n \rightarrow \infty}S_n = \frac{1}{4}$

Show that $\lim_{n \rightarrow \infty}S_n = \frac{1}{4}$ $$S_n = \sum_{k=1}^{n}\left(\sqrt{1 + \frac{k}{n^2}} - 1\right)$$ $$\sum_{k=1}^{n}\left(\sqrt{1 + \frac{k}{n^2}} - 1\right) < ...
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2answers
26 views

For a series, can you always find a subseries whose sum is smaller in magnitude?

Let's say you got a series $a_n$, such that $\sum^\infty_{n=0}a_n=L>0$. For any $0 \lt K \le L$, can you always find a subsequence $b_n$ of $a_n$, such that $\sum^\infty_{n=0}b_n=K$? If not always, ...
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12answers
2k views

Can an infinite sum of irrational numbers be rational?

Let $S = \sum_ {k=1}^\infty a_k $ where each $a_k$ is positive and irrational. Is it possible for $S$ to be rational, considering the additional restriction that non of the $a_k$'s is a linear ...
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0answers
20 views

Closed form expression for the constrained least squares problem

$\underset{A}{\min} \hspace{15mm} \sum_{k=1}^{N-1} ||\textbf{x}_{k+1}-A\textbf{x}_{k}||^2_2$ $s.t \hspace{15mm} W.A = 0 $ where (.) is the element by element product and $W$, $\textbf{x}_{i}$ are ...
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1answer
39 views

Prove a vector in $\ell^2(\mathbb{Z})$ is zero

Suupose we take a vector $\vec{c}\in\ell^2(\mathbb{Z})$ where $$c(i)=\sum_{k=1}^\infty\frac{c(-k+i)+c(k+i)}{k+1}$$ That is, every elements of the vector is a series with the other terms in $\vec{c}$. ...
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5answers
63 views

Summation for $\sum\limits^5_{i=2}\:\left(3i\:-\:5\right)$

I know that the closed form of $\sum\limits^n_{k=1}\:k=\frac{n(n+1)}{2}$ But I'm not sure what the closed form for $\sum\limits^5_{i=2}\:\left(3i\:-\:5\right)$ would be. Any push in the right ...
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2answers
33 views

Simple expression for $\sum_{k=1}^{n-1}\:\frac{1}{k\left(k+1\right)}$

I know that $\:\:\frac{1}{k\left(k+1\right)}\:\:\:\:=\:\frac{1}{k}\:-\:\frac{1}{k+1}\:$ And that $\sum_{k=1}^{n-1}\:k$ $= \frac{n(n-1)}{2}$ But I'm not completely sure how to turn ...
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1answer
112 views

About the sums $\sum_{n=1}^\infty x^{n^2}$ and $\sum_{n=1}^\infty \frac{x^n}{1+x^{2n}}$

Despite all my efforts trying to crack these, i haven't been able to do so. An approach that i've tried gives me somewhat of an asymptotic approximation, but still fails to produce the values near ...
1
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1answer
21 views

Summation of series with binomial coefficients

The value of $$\sum {n\choose n-r} (n-r) \sin(r\cdot \pi/n)$$ where $r\in (0 ..,n)$ is equal to? I think the question can be solved by writing the series in reverse order but I am not able to solve ...
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0answers
49 views

Differentiate a geometric sum and show that it is less than an equation

The question: a) Differentiate both sides of the geometric series with respect to $r$: $$~~\displaystyle\sum_{i=0}^nr^i=\frac{1-r^{n+1}}{1-r}$$ b) Use the result in part (a) to show that (Assume ...
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2answers
55 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 ...
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0answers
19 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 ...
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0answers
67 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 ...
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3answers
40 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 ...
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2answers
69 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!!
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1answer
40 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: $ ...
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2answers
114 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 ...
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5answers
68 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 ...
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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 ...
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1answer
90 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
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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, ...
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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
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3answers
40 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 ...
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0answers
27 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} ...
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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
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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 ...
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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 ...
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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 ...
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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 ...
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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} ...
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1answer
24 views
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0answers
91 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 ...
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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 ...
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1answer
42 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
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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: ...
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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: $1+2+3+4+5=15$ $7+8=15$ $4+5+6=15$
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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 ...
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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 ...
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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 ...
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1answer
43 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 ...
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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 ...
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1answer
33 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 ...
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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
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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 $
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1answer
28 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
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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 ...
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2answers
45 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} ...
205
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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
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3answers
46 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: $$\sum_{i=1}^{n-1}\sum_{j=i+1}^n i+1$$ ...