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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2answers
104 views

Find : $-\frac{1}{2} +\frac{1}{3}-\frac{1}{4}+\cdots-\frac{1}{100}$

Find following sum : $$ -\frac{1}{2} +\frac{1}{3}-\frac{1}{4}+\cdots-\frac{1}{100}$$ I think we can write $$\frac{-1}{101}+\sum_{n=1} ^{50} \left( - \frac{1}{2n} +\frac{1}{2n+1}\right)$$
1
vote
2answers
67 views

why double Sigma summation?

The lecture slides is about covariance. One page it is on the other page, it is Why there are double Sigma summation on the first one?Is it just a typo?
3
votes
1answer
44 views

Closed form for $\sum_{p\le n, p\text { prime}}\frac {(-n)^p} {p^n}$

I'm looking for a closed form for the following sum $$\sum_{p\le n, p\text { prime}}\frac {(-n)^p} {p^n}$$ Motivation: This sum is part of a EXP-calculation formula in a game I'm helping develope ...
1
vote
0answers
52 views

Does the limit of the sum equal the sum of the limits in this case?

I'm interested in evaluating the following limit: \begin{align*} \lim_{N\to\infty}\sum_{n= 1}^N\frac{1}{N\sin\left(\frac{\pi n}{N}\right)} \end{align*} Because \begin{align*} ...
3
votes
0answers
108 views

What is asymptotics of this oscillatory double sum? (Fractal Dimension problem)

The term Gibbs Phenomenon refers to the peculiar way Fourier Series behave at sharp changes in a function's value. However, this problem becomes particularly annoying to deal with when trying to ...
9
votes
9answers
17k views

What's the formula for the 365 day penny challenge?

You might have seen the viral posts about "save a penny a day for a year and make $667.95!" The mathematicians here already get the concept while some others may be going, "what"? Of course, what the ...
0
votes
0answers
26 views

Average Sum bigger than average product?

How can I prove that $\sum_{k=1}^K p_kt_k > \prod_{k=1}^K t_k^{p_k}$ given that $t_k>0$, $0<p_k<1$ and $\sum p_k = 1$ Thank you
0
votes
4answers
66 views

Do summations always increment by one? How do you incrment by a negative number? Or any real number?

Do summations always increment by one? Having a more of background in programming than math. I am just learning about summations and I look at them as loops that increment by one. If my assumption ...
0
votes
2answers
73 views

Could we solve $\int_{0}^{\infty}\sin(x)dx$ and what does it say about $\lim_{x\to\infty}\cos(x)$?

As the title states: Could we solve $\int_{0}^{\infty}\sin(x)dx$ and what does it say about $\lim_{x\to\infty}\cos(x)$? It is clear we can't solve this using the fundamental theorem of Calculus, but ...
3
votes
4answers
103 views

What is $\sum_ {i=1}^{10}2$?

What is $\displaystyle\sum\limits_{i=1}^{10}2$? Is it $$\sum\limits_{i=1}^{10}2 = 200$$ or $$\sum\limits_{i=1}^{10}2 = 2$$ or $$\sum\limits_{i=1}^{10}2= 0$$ or none of the above? I feel a ...
1
vote
2answers
40 views

For all $n >0$, $ \left(1+\frac{1}{n} \right)^n = 1+ \sum_{k=1}^n\bigl[ \frac{1}{k!} \prod_{r=0}^{k-1}(1-\frac{r}{n}) \bigr]$ [duplicate]

I am working through some problems on induction and I have been stuck on this one for a while. If anyone has any hints. I can show it is true for the $n=1$ and $n=2$ case but I am having difficulty on ...
2
votes
0answers
39 views

Finding a summation involving gcd

I am trying to evaluate the following sum: $$b\sum_{i=a}^b\frac{i}{\gcd(i,b)}$$ I have solved the problem if $a=1$ but I am clueless for the case when $a$ is not $1$. For $a=1$, I used the fact that ...
0
votes
1answer
18 views

Computing the conditional expectation of $E[t^Y|X=k]$?

For $X\in $Bin($n,p$) and $Y|X=k \in$Bin($k,p$), I have the conditional expectation, $$ E[t^Y|X=k]=\dots $$ I know that I probably should use the formula; $$ E[Y|X]=\sum_{y}y \cdot f(y|x), $$ But how ...
4
votes
1answer
76 views

Finding the infinite series: $\sum_{m=0}^\infty \sum_{n=0}^\infty\frac{m!\,n!}{(m+n+2)!}$

Evaluate $$\sum_{m=0}^\infty \sum_{n=0}^\infty\frac{m!n!}{(m+n+2)!}$$ involving binomial coefficients. My attempt $\frac{1}{(m+1)(n+1)}\sum_{m=0}^\infty ...
3
votes
1answer
50 views

How to find the approximation of a series with the help of Riemann sum? [duplicate]

Given a series: $$\frac{1}n+\frac{1}{n+1}+\cdots+\frac{1}{2n-1}$$ What are these types of questions called and what is the strategy for them? The next step in the solution manual is: ...
1
vote
0answers
39 views

How do I determine convergence of the following sum using Cauchy test?

The sum is $ \sum _{n=1}^{\infty }2^n\left(\frac{n}{n+1}\right)^{n^2}\: $ I started by studying the asymptotic behavior $ ...
0
votes
0answers
18 views

Opposite of Monte Carlo

In this lecture, at 1:08:35, the lecturer goes from $$\text{argmin}\frac{1}{N}\sum\limits_{i=1}^{N}\text{log}\frac{p(x_i|\theta_0)}{p(x_i|\theta)}$$ to ...
2
votes
2answers
261 views

Verify If Sum of Factorials is Divisible by Integer

I am working on preparing for JEE and was working on this math problem. We have the sum, $$\sum_{n=1}^{120}n!=1!+2!+3!+\ldots+120!$$ Now I am given the question, which says that what happens when ...
0
votes
0answers
87 views

Difficult expression of sum

I wanna show that $\sum{\frac{1}{k^{4}}}=\frac{\pi^{4}}{90}$. For this, I know that $$\sin(z)=z-\dfrac{z^{3}}{3!}+\dfrac{z^{5}}{5!}-\dfrac{z^{7}}{7!}+\cdots$$ On the other hand, also know that ...
3
votes
1answer
61 views

Newton's Sums might be helpful but a bit long.

I tried this question with basic Newton's Sums approach but it was kinda long. Can anyone post a bit smaller and logical method ? Consider the polynomial $$P(x)=6x^5+5x^4+4x^3+3x^2+2x+1$$ Given ...
1
vote
2answers
50 views

Evaluate an infinite sum

I've been trying to find a way to evaluate a sum and i can't. I lost some classes and now find it difficult to understand, the notes that i've been given are not specific and i've been googling for ...
3
votes
4answers
73 views

The value of series $\sum_{n=1}^{\infty}\frac{n}{2^n}$ [duplicate]

The value of series $\sum_{n=1}^{\infty}\frac{n}{2^n}$ I try to write some terms,but of no use. Is there any general method to approach such questions. Thanks
2
votes
3answers
74 views

How do I evaluate the sum $\sum_{k=1}^\infty\left(\ln\big(1+\frac{1}{k+a}\right)-\ln\left(1+\frac{1}{k+b}\big)\right)$ [closed]

How do I evaluate the sum $$\sum_{k=1}^\infty\left(\ln\Big(1+\frac{1}{k+a} \Big)-\ln\Big(1+\frac{1}{k+b}\Big)\right)$$ where $0 <a<b<1$? Hints will be appreciated Thanks
1
vote
2answers
37 views

Square of a Sequence of Numbers

This is a simple question, but I could not find the solution. What is the compact form of expansion of $(n_1+n_2+\ldots+n_k)^2$ Is it: $\sum_{i=1}^k n_i^2 + 2\sum_{i=1}^{n-1} ...
0
votes
0answers
15 views

Show that it is a positive summation kernel

Let $$ K(x) = \frac{3}{4}(1-x^2) $$ for $|x|<1$ and $0$ elsewhere Show that $$ K_n(x):=nK(nx) $$ is a positive summation kernel. Solution: So I want to show that $$ K_n(x)≥0 \\ ...
1
vote
1answer
16 views

Help in explaining solution to sequence problem (finite sum)

The problem is as follows: A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and ...
1
vote
1answer
107 views

How can I find the exact value of the $\sum\limits_{i=1}^\infty\frac{x^i}{i}$? [closed]

I tried to solve with derivatives and integrals but this way I can only find that its closed form is $-\ln(1-x)$. How to find the exact value ?
15
votes
1answer
318 views

New Year Summation 2016: $\displaystyle\sum_{r=3}^{\; 3^2}r^3$

Decode the following summation to welcome the new year! Find integer $n$ such that $$\large\color{darkblue}{\sum_{\qquad \qquad r={\sum_{m=0}^\infty\left(\frac{n-1}n\right)^m }}^{\qquad \qquad ...
3
votes
2answers
46 views

Find a relation between the sum of $100$ positive numbers and the sum of their pairwise products

For positibe real numbers $a_1,a_2,\dots,a_{100}$, let $$p=\sum_{i=1}^{100}a_i \quad \text{and} \quad q=\sum_{1\le i<j \le 100} a_i a_j$$ Then (A) $q=\dfrac{p^2}2$ (B) $q^2\ge\dfrac{p^2}2$ ...
8
votes
0answers
165 views

A sum that holds an apparently “surprising” closed form.

$$\large{\sum_{n=1}^{\infty}\left( 1-\dfrac{\log (n^3+1)}{\log (n^3+n+1)}\right)}$$ I got this series from my friend, who stated this holds a "surprising" closed form. Could I have some help ...
1
vote
1answer
25 views

Make explicit formula of a combined sum

I need to get the sum of the next series: $\sum_n^m(n*(\sum_n^mn)) \Rightarrow [n,m]\in \Bbb{N}$ So i am not able to make it explicit. But i can't combine the $\sum_n^m = \frac{n(n+m)}{2}$ formula ...
12
votes
2answers
269 views

Is there a closed form expression for the “generalized” addition of the first $n$ numbers?

Firstly, I will explain what I am trying to do intuitively. We take the sum of the first $n$ positive integers. Let's say this sum is equal to $q$. Then you add that sum to the sum of the first $q$ ...
0
votes
1answer
28 views

Notation for a specific summation

I'm trying to make the sum stop before the summation has a negative exponent. For example, I would want the sum to stop at $2^0$ in $2^3+2^2+2^1+2^0+2^{-1}$ The sum I'm dealing with is $$\sum ...
0
votes
1answer
28 views

What is summation notation for functions of decreasing integers?

Wondering how may write following expression in sigma notation for summation? \begin{eqnarray} S &=& f(x_1,x_2,\cdots,x_n-1)+f(x_1,x_2,\cdots,x_n-2)+\cdots+f(x_1,x_2,\cdots,0)\\ ...
0
votes
1answer
61 views

Estimating $\sum\limits _{n=1}^k \sin \frac x n$ in the form $f(k,x) \sin(g(k,x))$

When you plot the function for a reasonably large $k$ ($300$ in this case) you get something like this... This seemed like it could be estimated the way I stated previously. The accuracy of that ...
0
votes
1answer
127 views

Calculating the value of $\sum_{n=0}^{\infty}\frac{1}{n!(n+1)!}$ [closed]

What's the value of: $$\sum_{n=0}^{\infty}\frac{1}{n!(n+1)!}$$
2
votes
0answers
93 views

Sum of $\sum\frac{1}{i^{i}}$

Few minutes ago I saw the sum: $$\sum(i^{i})$$, and there is no normal equation to describe that. My question is: Find the $$\sum\frac{1}{i^{i}}$$ , if it's exist of course.
2
votes
1answer
102 views

Finding $f(x)$ such that $\int_{a}^{b}f(x)dx=\sum_{k=a}^{b}f(k)$

Does there exist any method to find the function $f(x)$ which satisfies $$\int_{a}^{b}f(x)dx=\sum_{k=a}^{b}f(k)$$ For example $$\int_{- ...
0
votes
1answer
29 views

Simplifying a -1 term out of a finite product

I've come up with an algorithm that relies upon the value of the following product: $$Q_{k} =\prod_{n=0}^k [f(n) - 1]$$ Where $f(n) \ge 2$ and strictly increasing integer function [see note]. ...
2
votes
3answers
50 views

Sum with power in numerator with alternating signs and factorial in the denominator

$$\sum _{ k=0 }^{ 50 }{ \frac { { (-1) }^{ k+1 }{ 3 }^{ k } }{ (2k)!(100-2k)! } } $$ I guessed the summation above equals to $\frac { { 2 }^{ 99 } }{ 100! }$ and it turns out to be true after I ...
0
votes
2answers
25 views

General summation formula for…

I'm looking for a general summation formula for $\left((1-\lambda)\lambda^0+(1-\lambda)\lambda^1\right)R^{(2)} + \left((1-\lambda)\lambda^2+(1-\lambda)\lambda^3\right)R^{(4)} + ...
0
votes
0answers
50 views

How prove that $\sum_{k=1}^n \lfloor (\frac{k}{2})^2 \rfloor =\lfloor \frac{1}{24}n(n+2)(2n-1) \rfloor$

How prove that $\sum_{k=1}^n \lfloor (\frac{k}{2})^2 \rfloor =\lfloor \frac{1}{24}n(n+2)(2n-1) \rfloor$? What is $\sum_{k=1}^n \lfloor (\frac{k}{3})^3 \rfloor$?
1
vote
1answer
59 views

How find $\sum \limits _{k=1}^n \frac{1}{(k+1) \sqrt{k} + k \sqrt{k+1} }$

How find sum $\sum \limits _{k=1}^n \frac{1}{(k+1) \sqrt{k} + k \sqrt{k+1} }$ ? Maybe there is a simple way.
1
vote
2answers
115 views

How to find 10th digit of $\sum_{k=1}^{49} k!$

How to find the tenth digit (from the right) of $\sum_{k=1}^{49} (k!)$ analytically. The worst possible method would be to actually sum each individual number which would yield a number of order ...
9
votes
2answers
219 views

Contest math problem: $\sum_{n=1}^\infty \frac{\{H_n\}}{n^2}$

$$\sum_{n=1}^\infty \frac{\{H_n\}}{n^2}$$ I have managed to prove that it converges, but am having trouble with a closed form. This came from a school contest from last year, but can't really figure ...
0
votes
0answers
76 views

How prove $\frac{1}{\sin^2(\frac{\pi}{2n})}+ \frac{1}{\sin^2(\frac{2\pi}{2n})}+ \cdots + \frac{1}{\sin^2(\frac{(n-1)\pi}{2n})} =\frac{2}{3}(n-1)(n+1)$ [duplicate]

How prove $$\frac{1}{\sin^2(\frac{\pi}{2n})}+ \frac{1}{\sin^2(\frac{2\pi}{2n})}+ \cdots + \frac{1}{\sin^2(\frac{(n-1)\pi}{2n})} =\frac{2}{3}(n-1)(n+1) \text{ ?}$$ Maby a simple way?
3
votes
2answers
65 views

Expression of the sum of a series

I am unable to calculate the expression of the sum of the series $1^{3/2} + 2^{3/2} + \cdots + n^{3/2}$. Could you please help me finding the answer.
5
votes
5answers
91 views

Prove that $\sum_{k=0}^n(-1)^k\binom{n}{k}\binom{m+k}{r}=0$

It is well-known that $\sum_{k=0}^n(-1)^k\binom{n}{k}=(1-1)^n=0$. It is seems like that $$\sum_{k=0}^n(-1)^k\binom{n}{k}\binom{m+k}{r}=0$$ for any $m,r\in\mathbb{N}$, $r\leq m$. How to prove or ...
0
votes
1answer
93 views

Formula to calculate directly $ 1 + 2 \cdot 3 + 3 \cdot 4 \cdot 5 + 4 \cdot 5 \cdot 6 \cdot 7 + \dots + n \cdot (n+1) \cdot \dots \ (2n-1)$

Is it some formula to calculate $$ 1 + 2 \cdot 3 + 3 \cdot 4 \cdot 5 + 4 \cdot 5 \cdot 6 \cdot 7 + \dots + n \cdot (n+1) \cdot \dots \ (2n-1)$$ for a given $n$ without iteration? comes from : ...
1
vote
0answers
67 views

Show crazy identity with too many sums in it (for numbers $\tau_1,…, \tau_{N-1}$). [closed]

Let $N \geq 2$ be a natural number and $ i \in \{1,\ldots,N-1 \}$. Then we define the number $\tau_i$ via $$ \tau_1 := \frac{1}{N} \sum_{k=1}^{N-1} \sum_{l=1}^k \frac{N^2}{l(N-l)}, $$ $$ \tau_i := ...