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
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.
12
votes
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 ...
0
votes
1answer
37 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
votes
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
votes
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 ...
-3
votes
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
vote
1answer
49 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 ...
1
vote
3answers
221 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
0
votes
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 ...
1
vote
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
25 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 ...
0
votes
1answer
61 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
vote
1answer
27 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 ...
1
vote
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
votes
1answer
37 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 $n=0$}\\ ...
1
vote
0answers
43 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
votes
2answers
83 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.
1
vote
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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
votes
1answer
65 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 ...
0
votes
2answers
125 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
106 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
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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?
1
vote
2answers
58 views

How to prove this identity [closed]

I would like to prove the following identity without using induction: $$\sum _{ k=1 }^{ n }{ { (-1) }^{ k } {n\choose k} }\cdot k^n=(-1)^n\cdot n!. $$
8
votes
2answers
215 views

How prove this sum $1+\sum_{n=1}^{\infty}(1+x^n)(\frac{(1-y)(1-yx)(1-yx^2)\cdots(1-yx^{n-1})}{(y-x)(y-x^2)(y-x^3)\cdots(y-x^n)}=0$

let $|x|<1,|y|>1$, show that $$1+\sum_{n=1}^{\infty}\left((1+x^n)\left(\dfrac{(1-y)(1-yx)(1-yx^2)\cdots(1-yx^{n-1})}{(y-x)(y-x^2)(y-x^3)\cdots(y-x^n)}\right)\right)=0$$ by this sum,I ...
8
votes
4answers
209 views

Does $\sum_{k=0}^{k=n} {n \choose k} k!$ have a closed form for integers $k,n$?

While doing research in computer system, I came across the following summation: $$S_n = \sum_{k=0}^{n} {n \choose k} k! = \sum_{k=0}^{n} \frac{n!}{(n-k)!}$$ where both $n$ and $k$ are integers. $S_n$ ...
0
votes
0answers
37 views

Are generating functions ever analytic for logarithmic series?

Given a series $s_n = \ln(n) f(n)$ where $f(\cdot)$ is an elementary analytic function which does not involve the logarithm. More precisely $f$ can have simple poles but no branch cuts or essential ...
1
vote
1answer
102 views

What's it equal to: $\lim_{n\rightarrow \infty }\sum_{1\leqslant k\leqslant n}\frac{1}{k\ln (n+k)}$

What's it equal to: $$\lim_{n\rightarrow \infty }\sum_{1\leqslant k\leqslant n}\frac{1}{k\ln (n+k)}$$
1
vote
2answers
77 views

Find $\sum\frac{a(n)}{n(n+1)}$, where $a(n)$ — number of 1's in binary expansion of n. [duplicate]

Let $a(n)$ is a number of 1's in binary expansion of n, find the sum $$ \sum\limits_{n=1}^{\infty}\frac{a(n)}{n(n+1)}. $$
1
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4answers
87 views

Finding the sum of series $\displaystyle\sum \limits_{n=1}^{\infty} (-1)^{n}\frac{n^2}{2^n}$

I have some problems in finding the values of series that follow this pattern: $$\sum \limits_{n=0}^{\infty} (-1)^{n}*..$$ For example: I have to find the value of this series $$\sum ...
0
votes
1answer
18 views

Directional Derivate for sums

I know the directional derivative as $D_uf(x)=\nabla f(x) . u$ But I do not know how this applies here?
5
votes
1answer
124 views

Summing a series

This problem was inspired by a typo on a homework assignment for Calculus 2, which covers integration and series. Find the sum of $$\sum_{n=0}^{\infty} \frac{1}{2^{n^2}}$$ Does anyone have any idea ...
1
vote
1answer
35 views

How to evaluate a limit that involves matrices

I've stumbled upon this problem while I was browsing through the contents of an admission exam . I've struggled tremendously with this exercise and I've got no idea what do to next , it's eating me ...
1
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0answers
47 views

Infinite sum with infinite poles

I want to evaluate the sums $$ \sum_{q=-\infty}^\infty\frac{1}{\sin\left(a q + x\right)\left(q + b \right)^4} $$ and $$ \sum_{q=-\infty}^\infty\frac{\cos\left(a q + x\right)}{\sin^2\left(a q + ...
2
votes
3answers
97 views

Summation with Ceilinged Logarithmic Function

According to Johann Blieberger's paper - "Discrete Loops and Worst Case Performance" (1994): $$ \sum_{i = 1}^{n}\left \lceil \log_2{(i)} \right \rceil = n\left \lceil \log_2{(n)} \right \rceil - ...
0
votes
1answer
10 views

Sigma sums problem

Could you please help me with the below result? How do we get from here $(1+2\sum_{j=1}^\infty\rho^j)π_0=1$ To here $π_0=(1-\rho)/(1+\rho)$ Many thanks
3
votes
0answers
58 views

Evaluation of the double sum

Is there a way to get a closed expression for the double sum: $$\sum\limits_{n = 1}^\infty \sum\limits_{m = 1}^\infty \left( \frac{m}{(n^2 + m^2 - E_1)(n^2 + m^2 - E_2)} \right)^2$$ , where $E_1$ and ...
1
vote
1answer
83 views

How to show $\sum_{n=1}^{\infty}\frac{H_{n}}{n^{2}}=2\zeta (3)$? [duplicate]

How to show this equation is true. $$\sum_{n=1}^{\infty}\frac{H_{n}}{n^{2}}=2\zeta (3)$$ where $H_{n}=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$
2
votes
0answers
30 views

Inverse logarithmic integral

If the expansion of the logarithmic interval is$$\text{li}(n) = \log \log n + \gamma + \sum_{k=1}^\infty \dfrac{(\log n)^k}{k! k}$$ what is the inverse of the function?
0
votes
1answer
40 views

How can I derive this summation?

I have the following equation, $$ K_r=\left ( \frac{P}{RT} \right )^{v}exp \left \{ \sum_{s}\left [ (\beta_{s,r}-\alpha_{s,r}) \left \langle \frac{h_s}{RT}-\frac{s_s}{R}\right \rangle \right ] ...
2
votes
8answers
187 views

How to find the sum: $1^{\frac{1}{3}}+2^{\frac{1}{3}}+3^{\frac{1}{3}}+ . . . +(50)^{\frac{1}{3}}$

Can some one help me to find the sum of the following expression? $$1^{\frac{1}{3}}+2^{\frac{1}{3}}+3^{\frac{1}{3}}+ . . . +(50)^{\frac{1}{3}}$$
0
votes
2answers
36 views

Calculate simple non geometric sum

I want to calculate this sum. How can I do it ? $$ \sum_{i=1}^n i \cdot \left(\frac{9}{10}\right)^i$$ I know how to calc geometric sum, but how to calc this?