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

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1answer
20 views

Interchange summation (outer infinite, inner dependent on outer)

For finite summation limits, I believe that the following holds (for some general function $f$): $\sum_{i=2}^n \sum_{j=1}^{i-1} f(i,j) = \sum_{j=1}^{n-1} \sum_{i=j+1}^{n} f(i,j)$ ... (1) However, I'm ...
1
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0answers
27 views

Derive an inequality using Summation by Parts

Can someone help me to derive the following inequality using Summation by Parts? $a_n$ is a decreasing sequence of positive terms. $$\left|\sum_{k=m+1}^{n+p} a_k \sin kx\right| \le ...
-4
votes
1answer
33 views

Solving mathematical summation [on hold]

How to solve the following sum: $$\sum_{i = 0}^n \frac{(1/2^n)^i \cdot (1 - 1/2^n)^{n-i}}{i!\cdot(n-i)!}$$
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0answers
24 views

Find minimum value of $n$ for an integer $A$ such that $A=n^x$,where $n>1$ and $x\geq 1$

How can I calculate sum of a series of function $f(A)$ for $A = 2,3,4,5,6...A$ $f(A)=n$ (such that $n$ is minimum integer such that $A=n^x$ where $n>1$ and $x≥1$ and both n and x are integer) ...
7
votes
3answers
397 views

Summation of a term to infinity

I read through many tutorials but no one mentioned this explicitly. Is the following conversion valid? $$\sum_{k=0}^\infty \frac{k-1}{2^k} = \lim_{n\to \infty} \sum_{k=0}^n \frac{k-1}{2^k}$$ ...
0
votes
0answers
32 views

The partial sums of the form $\zeta _N(s)= \sum_{k=1}^{N} \frac{1}{k^s} $

My question concerns the partial sums of the form: $$\zeta _N(s)= \sum_{k=1}^{N} \frac{1}{k^s}$$ Is there an analytic continuation to the entire complex plane such sums as the complex functions of s ...
4
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1answer
20 views

What about $\sum_{\substack{2\leq n\leq y,\text{n prime}}}n\log\log n$ when $y=[x]\to\infty$?

For a real $x\geq 2$ and when we take $y= [x]$ its integer part, I am trying to study the asymptotic size or growth of $$\sum_{\substack{2\leq n\leq y,\text{n prime}}}n\log\log n,$$ I believe that ...
0
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1answer
46 views

Summation Problem?

It's been a while since I've had to do anything like this so my skills are super rusty. Someone at the office asked me if I knew a formula and I came back empty. I have $115$ numbers ($n$) that sum ...
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2answers
74 views

Can sum of rationals be irrational? [duplicate]

I have seen proof of -: Sum of two rationals is rational and that implies that sum of all rational is always rational( by induction). Now my question is about Basel ...
0
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1answer
44 views

Way to calculate cumulative sum of a sequence m times without summing all elements m times.

Suppose we have a sequence 2 1 3 1 Now , I want calculate it's cumulative sum m times and determine the element at position x in the sequence. Lets's say I want to perform cumulative sum operations ...
0
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1answer
43 views

Finding closed form expression for a multiple sum.

Let $n_1$, $n_2$ and $m$ be non-negative integers and let $\theta_1$ and $\theta_2$ be real numbers subject to $\frac{\theta_1}{\theta_2} = 1+m$. We consider a following multiple sum: \begin{eqnarray} ...
1
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1answer
26 views

Einstein's summation convention and double indices

So I'm actually rather familiar with Einstein' summation notation and I understand objects like $a^{\mu \nu} a_{\mu \nu}$ just fine. But now I'm suddenly wondering why I've never come across objects ...
2
votes
3answers
121 views

How to calculate $ \sum_{n=1}^{15}n(n!) = ? $

In a contest between me and my friend, i was able to solve all the questions till he stumped me at this one. $ \sum_{n=1}^{15}n(n!) =?$ The only thing I could think of how to pursue is $n(n+1)! ...
2
votes
1answer
37 views

Elementary differentiation problem involving logarithms: What am i missing here?

Consider the finite sum $S=$ $\sum_{k=2}^n \log k - \log(k-1)$. Differentiating $S$ w.r.t $k$, we have $S'= \sum_{k=1}^n \dfrac{1}{k} - \dfrac{1}{k-1}=-\sum_{k=1}^n \dfrac{1}{k(k-1)}<0$. But ...
-1
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2answers
110 views

What does it mean to say “again” or “finally” in math? [on hold]

According to math rules if we say again, does it mean we are saying repeat previous step? For example: You have $10$ coins. Add $10$ more coins. Add $2$ coins Again $2$. Again $2$. Add $2$. And ...
2
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4answers
20 views

Summation of series involving complex exponents.

I am interested to find out what is $\sum_{k=0}^{200} i^k$, where $i$ is complex number
5
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1answer
91 views

Why isn't finite calculus more popular?

I'm reading through Concrete Math, and learning about finite calculus. I've never heard of it anywhere else, and a Google search found very few relevant sources. It seems to me an incredibly powerful ...
3
votes
2answers
47 views

Show that the integer nearest to $\frac{n!}{e}$ $(n \geq 2)$ is divisible by $n − 1$ but not by $n$.

Show that the integer nearest to $\frac{n!}{e}$ $(n \geq 2)$ is divisible by $n − 1$ but not by $n$. I am still trying to improve my basic math skills but on this one i did not get far. Taylor ...
1
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1answer
30 views

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 ...
0
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1answer
31 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
46 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, ...
57
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17answers
7k 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 ...
3
votes
1answer
41 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 $\vec{c}$. ...
-1
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4answers
69 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 ...
1
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2answers
35 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 ...
4
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0answers
118 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
23 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 ...
0
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0answers
53 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 ...
2
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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 ...
3
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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 ...
6
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0answers
71 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 ...
1
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3answers
41 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 ...
1
vote
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!!
1
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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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votes
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 ...
1
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5answers
74 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
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2answers
49 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 ...
5
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1answer
91 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
44 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
42 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
29 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} ...
0
votes
1answer
22 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
68 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 ...
2
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1answer
51 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 ...
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 ...
0
votes
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
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1answer
24 views
9
votes
0answers
93 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 ...