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

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1answer
42 views

Prove that $\sum_{k=-\infty}^\infty e^{-j2\pi f k T}=\sum_{k=-\infty}^\infty\delta(f-\frac{k}{T})$

This is part of a proof itself. $\sum_{k=-\infty}^\infty e^{-j2\pi f k T}=\sum_{k=-\infty}^\infty\delta(f-\frac{k}{T})$ $\delta$ is Dirac function. It's been a while I am thinking about this part ...
0
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1answer
54 views

Find the set of convergence $\sum_{n=1}^{\infty} \frac{1+x^n}{1-x^n}$

How the interval [a, b]: $x \in [a,b]$ can be found for the next sum? $$\sum_{n=1}^{\infty} \frac{1+x^n}{1-x^n}$$ The sence to check the next limit $$\lim_{n \to \infty} \frac{1+x^n}{1-x^n} = ...
1
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1answer
27 views

$\sum^{\infty}_{n=1} \log(\frac{n+a+b}{n+a} \times \frac{n+b}{n+b+1})$

Prove $$\sum^{\infty}_{n=1} \log(\frac{n+a+1}{n+a} \times \frac{n+b}{n+b+1})=\log\frac{1+b}{1+a}$$ Hints/Answers are appreciated
2
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5answers
125 views

Find a closed form for $\sum_{k=0}^{n} k^3$ [duplicate]

Find a closed form for $\sum_{k=0}^{n} k^3$. I would appreciate ideas for approaching questions like this in general as well. Thanks.
0
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2answers
26 views

How should I solve this summation problem?

Lets say that we have these $x$ and $y$ coordinates $x=1,2,3,4,5$ and $y=6,7,8,9,10$ and where $n=5$. How would I use these $x$ coordinates with the first summation? Now, I know that learning is ...
1
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1answer
60 views

$\sum\limits_{i,j\in[|1,n|]}\frac{\max\left(i,j\right)}{\min\left(i,j\right)}$

I'm trying to do this sum $\sum\limits_{i,j\in[|1,n|]}\frac{\max\left(i,j\right)}{\min\left(i,j\right)}$ What I have tried : $\sum\limits_{i,j\in[|1,n|]}\frac{\max ...
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2answers
43 views

double summation problem $\sum^5_{i=1}i \times \sum^5_{j=1}j =…$ please check

(I) $\sum^5_{i=1}i \times \sum^5_{j=1}j = 1 \times (1) +1 \times (2) + \cdots +1\times (5) +2\times (1)+2\times (2) +\cdots + 2\times (5) + 3\times (1) + 3\times (2) + \cdots +3\times (5) + 4\times ...
0
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0answers
39 views

Prove that maximum 9 trailing zeroes in this summation

I am trying to prove that there are a maximum of 9 trailing $0$'s at the end of this summation: $$\sum_{k=1}^{k=m} k^n$$ for $1\le n\le 1000000$ and $m\le 100$. Any help on how to approach?
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3answers
24 views

Positive positive negative negative Series

What is the simplest series that alternates in order $+,+,-,-,+,+,-,- \dots$ Specifically I want to make a Riemann sum for something, but it has this reoccurent pattern I haven't previously ...
4
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1answer
117 views

$\sum\limits_{k=0}^{n}\binom{2n+1}{2k+1}2^{3k}$ isn't divisible by 5

I have no idea Prove that for any $n$ natural number this sum $$\sum\limits_{k=0}^{n}\binom{2n+1}{2k+1}2^{3k}$$ isn't divisible by $5$. $\begin{array}{l} \left( {1 + x} \right)^{2n + 1} - ...
2
votes
5answers
109 views

Finding the minimum value of a sum [closed]

Let $x,y,z$ be real numbers . Find the real number $a$ so that $S$ has a minimum value , where $$S=|x-a|+|y-a|+|z-a| .$$
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0answers
30 views

Finite geometric sequence with a ratio greater than 1

I am trying to solve the recurrence relation: $ t(n)= 3T(n/2)+Cn$ (if $n>2$) $t(2)=C$ (otherwise) I know that there is the Master Theorem but I am trying to use the tree method. This ...
1
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1answer
66 views

Formula to find the sum of nth row?

In the following triangle I need to find the sum of nth row. Is there a general formula for this? If yes, then please tell me. Triangle: Row 1: 1 Row 2: 1 2 1 Row 3: 1 3 6 3 1 Row ...
3
votes
1answer
76 views

Asymptotic of a sum evaluation as $ x \to \infty $

Let be the sum $$ \sum_{n\le x}[x/n]=g(x) $$ where $ [x] $ means floor function. My best try for asymptotic is $ g(x) \sim x\log (x)+\gamma x +1$ where I have used the asymptotic $ [x] \sim x $ ...
1
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1answer
46 views

Proof for $\sum_{x=1}^{n-1}\lfloor \dfrac{mx}{n}\rfloor=\dfrac{(n-1)(m-1)}{2}$ where $(m,n)=1$

This identity might be well-known, but I could find the proof neither by myself not by searching it in Internet. Could you describe an outline of solution?
0
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1answer
34 views

Summation formula in dimension 2

One of the most common tools in analytic number theory is the summation by parts, my question is what is the similar formula when we are, for example, in dimension two and we have the sum $$ ...
0
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0answers
34 views

What are the advantages/disadvantages of integration vs. summation?

If we are given a function, $f(x)$, we can either integrate it or sum it. I'm wondering what integration can do with $f(x)$ that summation can't, and what summation can do that integration can't. ...
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2answers
39 views

How to evaluate the following sum? $\sum_{i = 1}^n \left\lfloor \frac{3n-i}{2}\right\rfloor.$

What is the value of the following sum? $$\sum_{i = 1}^n \left\lfloor \dfrac{3n-i}{2}\right\rfloor.$$ Especially how to handle the sums with floors? This sum appeared while solving this problem. My ...
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3answers
61 views

How do I find the sum of the series?

$$\sum_{k=1}^{7}40 \left( \frac{1}{2}\right)^{k-1} = \frac{635}{8}$$ The image of the orginial eqn is on the link above and so is the answer, but I need help in how to solve it. when I did solve it I ...
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1answer
38 views

Can we possibly exchange summation and integration with negative values?

This is an attempt to go further than this answer. Essentially, we have either a summation of an integral: $$\sum_x{ \left( \int{ f(x)dx } \right) } \tag{1}$$ ...or an integral of a summation: ...
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1answer
39 views

Is it possible to get a formula for this summation

The binomial sum $$s_n=\binom{n}{0}+\binom{n+1}{1}+\binom{n+2}{2}+\cdots+\binom{2n}{n}$$ I tried solving through recurrence, but unable to find one.
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2answers
182 views

How to prove this series $\sum_{n=1}^{\infty}\dfrac{a_{n}}{(n+1)a_{n+1}}$ diverges

Question: Assume that $a_{n}>0,n\in N^{+}$, and that $$\sum_{n=1}^{\infty}a_{n}$$ is convergent. Show that $$\sum_{n=1}^{\infty}\dfrac{a_{n}}{(n+1)a_{n+1}}$$ is divergent? My idea: since ...
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1answer
36 views

Complex summation simplification

What I'm getting is $$\frac{( \sin (N+1)x - 2^N \sin x)}{(2^N(\sin x - 2))}$$ How do I simplify to the form they have given , please help. I hope it's clear because I don't know Ajax still ...
2
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1answer
63 views

When can we use substitution for both integrals and summations?

This question is partially inspired by Qiaochu Yuan's answer to "Will moving differentiation from inside, to outside an integral, change the result?". Essentially, I would like to know, if we have: ...
0
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1answer
33 views

Complex number summation.

$$ \sum_{n=1}^N\cos(2n-1)\theta=\dfrac{\sin(2N\theta)}{2\sin\theta}, $$ where $\sin\theta\neq0.$ Deduce that $$ \sum_{n=1}^N (2n-1)\sin\left[\dfrac {(2n-1)\pi}N\right]=-N\operatorname{cosec}\dfrac\pi ...
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3answers
56 views

Find value of x in the given expression [closed]

Find the value of x in the following expression 2^2 * 2^6 * 2^10 * ..... *2^x = (0.125)^-24
1
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1answer
48 views

Summation of cos (2n-1) theta

By considering $\sum\limits_{n=1}^N z^{2n-1}$, where $z=e^{i\theta},$ show that $$ \sum\limits_{n=1}^N \cos{(2n-1)} \theta = \frac{\sin(2N\theta)}{2\sin\theta}, $$ where $\sin\theta\neq0$ I ...
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3answers
216 views

Sum of digits of number from 1 to n

Is there any general formula for calculating the sum of digits of number from 1 to n? n < 10^9
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2answers
31 views

Evaluating a taylor series around a given point

So I'm having some trouble with the problem: Given that $\ln(x+1)=\sum_{n=1}^{\infty } \frac{(-1)^{n+1}}{n}x^{n}, -1<x\leq 1$, find the Taylor series of ln(x) around 3. For which x is this series ...
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3answers
66 views

Radius of convergence for a given sum

What is a brief description of the radius of convergence? How do you find the radius of convergence for $$\sum_{i=1}^{\infty}2^i\cdot x^{-3(i-1)}$$
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1answer
67 views

Understanding Recurrence Relation

as i ask question and answered by some Clever people at this topic: Recurrence Relation Solving Problem i try to learn new thing with new question very similar to get familiar with recurrence ...
2
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0answers
24 views

Interchanging index of summation in $d$ dimensions

Let $\alpha = (\alpha_{1}, \ldots, \alpha_{d}) \in \mathbb{Z}_{\geq 0}^{d}$ and let $|\alpha| = \alpha_{1} + \cdots + \alpha_{d}$. I have the following question about interchanging summations: Is ...
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1answer
59 views

Sum of all the positive integers problem [duplicate]

The staff of Numberphile has shown that the sum of all the integers from $0$ to $\infty$ is $-\frac1{12}$. Recently I was looking for the sum of all the (positive) integers from $0$ to $n$ and I found ...
1
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1answer
26 views

How to show $\sum_{d\mid k}\frac{\mu (d)}{d}\left(\log\left(\frac{x}{d}\right)+O(1)\right)=\left(\sum_{d\mid k}\frac{\mu (d)}{d}\right)\log x+O(1)$

How to show this is true. $$\sum_{d\mid k}\frac{\mu (d)}{d}\left(\log\left(\frac{x}{d}\right)+O(1)\right)=\left(\sum_{d\mid k}\frac{\mu (d)}{d}\right)\log x+O(1)$$ I'm studying the book which is ...
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0answers
55 views

Does $ \sum_{n=1}^{\infty} \left(\frac{\sin(n)}{n}\left ( \sum_{m=1}^{n}\frac{1}{m} \right )\right) $ converge? [duplicate]

I am trying to determine whether the $$ \sum_{n=1}^{\infty} \left(\frac{\sin(n)}{n}\left ( \sum_{m=1}^{n}\frac{1}{m} \right ) \right) $$ converges or not. I have tried the popular tests, but all ...
3
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1answer
36 views

$\sum_{k=0}^{n}(-1)^k {{m+1}\choose{k}}{{m+n-k}\choose{m}}$

I'm supopsed to show that if $m$ and $n$ are non-negative integers then $$\sum_{k=0}^{n}(-1)^k {{m+1}\choose{k}}{{m+n-k}\choose{m}} = \left\{ \begin{array}{l l} 1 & \quad \text{if $n=0$}\\ ...
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0answers
42 views

How find this sum $S=\sum_{i=1}^{m}(-1)^{a_{i}}\cdot 2^{m-i}$ and $2^i\equiv a_{i}\pmod n$

Question: let $n$ is give odd positive integer numbers,and $a_{i}\neq 1,0\le a_{i}\le n-1$, and $$2^i\equiv a_{i}\pmod n,i=1,2,\cdots,m-1$$ where $m(m\le n)$ such $2^m\equiv 1\pmod n$ ...
2
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2answers
82 views

How find this sum $S(x)=\sum_{k=1}^{\infty}\frac{\cos{(2kx\pi)}}{k}$

Find this sum $$S(x)=\sum_{k=1}^{\infty}\dfrac{\cos{(2kx\pi)}}{k},x\in R$$ my idea: since $$S'(x)=2x\pi\cdot\sum_{k=1}^{\infty}\sin{(2kx\pi)}$$ then I can't.
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1answer
15 views

Is the integral test not valid for negative, upwards-trending functions?

The integral test states that given a function $f(n)$ that is positive, continuous, and decreasing on the interval $x \geq 1$, and a series $a_n = f(n)$, $\int_1^\infty f(n)dn$ and ...
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2answers
50 views

Sum of potencies with higher potency as clue

I am supposed to calculate the following as simple as possible. Calcute: $$1 + 101 + 101^2 + 101^3 + 101^4 + 101^5 + 101^6 + 101^7$$ Tip: $$ 101^8 = 10828567056280801$$ I have absolutely no idea how ...
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6answers
59 views

Sum of eight even integers that cannot be repeated more than twice is $50$

The sum of eight positive even integers is $50$. If no integer can appear more than twice in the set, what is the greatest possible value of one of the integers? This was a question I encountered on ...
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1answer
28 views

Converting failure rates between periods

I'm trying to figure out how to convert an annual failure rate between periods. Assume failures are uniform and independent. I know that the quick, back-of-the-envelope way is simply to divide the ...
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0answers
12 views

Determining the parameters of a limit equation

Let an=(3n^3+2n^2+n+10)^(1/3) -an-b . Let A { (a,b)∈R2 | lim as n->infinity of an=1/9^(1/3) }. And I'm supposed to find S=Σ(a^3+27b^3) . My attempt : I've worked on the limit and got to this point ...
2
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1answer
59 views

simplifying a triple sum of products of binomial coefficients

Right now I have a horribly-looking triple sum ($x,y,z$ are non-negative integers and $x+y+z=N$): $$ W_{12}(x,y)=\frac{x}{N}\sum_{l=0}^{x-1}\sum_{l'=0}^{y}\sum_{l''=0}^{z}{x-1 \choose ...
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2answers
124 views

Recurrence Relation Solving Problem

Can anyone help me in solving this complex recurrence in detail? $T(n)=n + \sum\limits_{k-1}^n [T(n-k)+T(k)] $ $T(1) = 1$. We want to calculate order of T. I'm confused by using recursion tree ...
0
votes
1answer
39 views

What does it mean for a series to be convergent?

I have the definition: Let $(a_n)$ be a sequence of real numbers. Let $s_n=a_1+a_2+...+a_n$. We say the series $a_1+a_2+...$ is convergent if the sequence of partial sums $(s_n)$ is convergent. The ...
0
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0answers
10 views

Z- transform existence

under what circumstances does a function $ f(x) $ has a Zeta transform ?¿? is this enough that a) $ f(x) $ is continous and derivable b) $ f(x) \to 0 $ as $ x \to \infty $ or at least $ f(x) \to C ...
2
votes
1answer
104 views

Combinatorial proof involving reciprocals

This is a follow-up to this question: show that if $n$ is a positive integer then $$\sum_{k=1}^{n}\frac{(-1)^{k+1}}{k}\binom{n}{k} =\sum_{k=1}^{n}\frac{1}{k}\ .$$ I was able to answer the question by ...
0
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0answers
32 views

How to derive finite formula for PRODUCT of terms from 1 to n?

should i use limits? I totally forgot how to work with them, i can imagine doing summation(sigma) but not product
1
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3answers
37 views

Finding an exact solution to a difference equation

How would I solve an equation of the form: $u(n+1)=1/2u(n)+(1/3)^n$ when $u(0)=1$? I got an answer of the form $u(n)= c + \sum(1/3)^j*2^{j-1}$ but believe this is incorrect?